7. A, C, D 8. C 9. D 10. B 11. A. 1. true 2. false 3. true 4. false 5. false 6. false 7. JK * ) * ) * EF ) * JH ) 20. Sample: 21.

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2 hapter nswers ractice -.,..0000,.0000.,. -,.,..,.. ample: or. or. or 0. or. or G. Y or. an heagon. circle with equall spaced diameters. a. angle. handshakes. h =.. ample: he n(n ) farther out ou go, the closer the ratio gets to a number that is approimatel ,,,,,,, ractice Front Front Front Front Front Front ight ight ight ight ight op ight. Front.,, ractice -. * ). an two of the following:,,,,,,,. oints,, and are collinear.. es. es. no. no. es. no 0. es. es. es. * no ). es. es. G... the empt set 0. *K ) * ) M. M. ample: plane. ample: plane. * ). es. no. es. the empt set. no 0. es. es ractice -. true. false. true. false. false. false. JK *, ) HG * ). * H ) *. ) an three of the * ) following pairs: and ; and ; *HG ) * ) * ) * F ) * JH ) F * ) * GK ) * ) and J ; and * ) ; * and ) * ) ; * and ) * ) ; and *FG ) HG FK JK H JK FG J ; and ; and ; and ; and *KF ) J KG H KG JH ; and * ) * ) H * ) * ) FK JH G 0. planes and. planes and ; planes and. ) planes and. ) planes ) and ). ) ample: G ). ). ) F ) and or G and. F, F. GF, G. es 0. ample: U ight. ample: Front Q W op ight Front op ight Geometr hapter nswers

3 hapter nswers (continued) ractice or. - or. ; ; no. ; ; es 0. or = ; = ; =. = ; = 0; = 0. ractice -. an three of the following: &, &M, &M, &. &. &. & &, &, &F, &F, &F. &F, &. &, &. = ; ; ;. = ; ; ; 0 ractice X Y X X Y Z X Y.. K M M? X nswers Geometr hapter

4 hapter nswers (continued)..? X W ractice -. in.. 0 in... m. p. p..p. cm; cm. cm; cm. in.; in ft; ft. 0 m; m. 0 m; m. ;. ;. 0; p..p. 0,000p.. ; 0.. ; eteaching -., X. true. false. false. true. true. false 0. true ractice -... ".. " ".. (, ). (,). (0, -). (-,). (-0.,.). (, - ). (, -). (, 0) 0. (-, ). es; = = = =. " Q Z..... eteaching - a.,,,,, Front ight... (., ) Geometr hapter nswers

5 hapter nswers (continued) b.. a. Front op ight Front ight b. a. b. a. b.. Front Front Front ight ight Front op ight Front op ight ight Front op ight eteaching -.. heck students work.. Q. ample:, Q,.,, 0. ample: and *Q ) eteaching - amples: * ) * ) F * ) * ) * ). and. and. and * ) H. plane and plane FH. plane H and plane F * ). plane H, plane G, and plane. and *F ) G.. 0. Front op ight nswers Geometr hapter

6 hapter nswers (continued).... F. eteaching -. 0 km. km. 0 km. 0 km. 0 km. an. 00 km; = 00 eteaching - a. ach angle is 0. b. obtuse a. ach angle is. b. acute a. ach angle is 0. b. obtuse. 0, right., acute. 0, obtuse. 0, acute. 0, right. 00, obtuse eteaching - amples:... X Q. U K eteaching -.. XY 0., YZ., XZ =., eteaching - a. b 0, b 0, b 0, b, b, and b 0; heck students drawings. b.,,,,, and a. b, b, b, b and b ; heck students drawings. b. b. cm b cm. ample: X J V Z Y diameter units; radius units Y Z Geometr hapter nswers

7 hapter nswers (continued). ample: 0 units. ample: 0. units. he are close.. nrichment -. ; ;.. op ight side Front. ound ound of Games in ound otal Games laed 0. ottom eft side op ight side ack Front. he number in the fourth column is equal to the sum of the numbers in the third column up to the given row c + n = n nrichment Front op ottom op ottom ight ight side eft side ight side eft side Front ack Front ack.. ottom op ottom op ottom eft side ight side eft side ight side eft side ack Front ack Front ack 0. V = units ;.. = units. V = units ;.. = units. V = units ;.. = units. V = units ;.. = units. V = units ;.. = units. V = units ;.. = units. ample: he foundation drawings; ou can find the sum of the numbers on each square.. ample: When there are no valles in the cube arrangement, ou can count the total number of squares used in all si views of an orthographic projection to find the surface area. nswers Geometr hapter

8 hapter nswers (continued) nrichment -. o; the will be parallel.. o; the will be skew.. and ; and ; and... line l or. line b. line n or. line a 0. ; ;. and. and... heck students work.. heck students work.. or nrichment -. north south. east west. east west. north south. alwas. never. never. alwas. never ft. ample: If one airplane is fling east west and one is fling north south, then one was out of its proper altitude range. ample: If both are fling east west at the same cruising altitude, then the air traffic controller instructed one of the airplanes incorrectl.. First venue, econd venue, or hird venue. ark treet, First venue, econd venue, or Grove treet. econd venue or First venue. o; the streets are parallel.. arter treet, ak treet, Hill treet, ine treet, and rosstown oad nrichment -. mi. mi. mi., mi; F, mi. F; more shorter connections. I; onl one long connection. link with I; it is short and would increase the number of connections to I. nrichment III. II. I. units. units. unit nrichment F I F I. heagon F F and quadrilateral I 0. heagon and quadrilateral F I F F nrichment -.., 0..,.. H. ". "; heck students work.. trapezoid 0. triangle nrichment -.. ample:. nswers ma var... heck students work.. he are all the same, about.; p.. heck students work. arl mathematicians beliefs were incorrect as stated in ercise. hapter roject ctivit : aper Folding able: ; ; ; ; multipl b ctivit : reating heck students work. ctivit : Writing heck students work. G ctivit : esearching heck students work. F Geometr hapter nswers

9 hapter nswers (continued) heckpoint Quiz.,.,. G. n three points are coplanar.. intersecting. skew. parallel.., G,, and F 0. ample:,, F, and a. b. *F ) c. d. G. Yes; the are vertical angles.. o; there are no markings.. Yes; the are adjacent and m& + m& = 0.. o; ou cannot conclude this from the diagram. a. b. c ; acute. 0 ; obtuse. 0 ; right.. heckpoint Quiz... Front. 0. (, 0).. p ft. p ft hapter est, Form. dd ;,.. Multipl b - ; -,.. dd, then, then, and so on;,.. dd another side of a heagon; check students work.. he relationship ( - )( + ) = - is true for all whole numbers. o countereample eists.. ample: ft. ample: ft? l ft ight m 0 m. ossible answers:,, and ;,, and F;,, and (, -). ft; ft. cm; 0 cm. in.; in.. p ft; p ft. p cm;.p cm. p in.; p in. hapter est, Form. Multipl b ;,.. dd, add, add, add, and so on;,.. Insert one more dot; a circle with three dots, a circle with four dots ( ) ( ). he relationship = is true for all whole numbers. o countereample eists.. ample:. ample: cm cm cm 0 cm in. in.. M,,, and., M, and or M,, and. ample:, F,, and 0a. M 0b. M 0c. F 0d.. &. &. &. & and & or & and & a. b. c... Q Q nswers Geometr hapter

10 hapter nswers (continued) º; obtuse. º; acute.. tudent gives eplanations, descriptions, and a drawing that are generall clear but ma contain a few errors. tudent makes significant errors in eplanations and descriptions. 0 tudent makes little or no attempt. K : coring Guide a. ample:. G..... (, ). in.; in. 0. cm; cm. p in.; p in.. 0p cm; p cm lternative ssessment, Form K : coring Guide amples: a. terior ras form a right angle for each figure. he total number of ras doubles with each successive figure. wo parallel segments connect the innermost ras. b. ther answers are possible. For eample, instead of considering the total number of ras (,,, ), consider the ras in the interior of the right angle: 0,,,... In that case, ou can argue that comes net, giving a total of ras, not, in figure. c. same as in part b, ecept with ras (If considering the ras in the interior of the right angle, Figure will have 0 ras.) tudent gives a correct eplanation, clear descriptions, and an accurate drawing. b. o; an three points are coplanar. c. ample:,, and tudent gives an accurate diagram, correct answers, and a valid eplanation. tudent draws a diagram that contains some errors but gives answers to parts b and c that are correct. tudent draws a diagram that contains significant errors and gives answers to parts b and c that are incorrect. 0 tudent makes little or no attempt. K : coring Guide ample: (0 + mlq) tudent constructs the figures correctl. tudent constructs the figures, but the constructions ma contain some minor errors. tudent misses some ke ideas, resulting in significant errors in the constructions. 0 tudent makes little or no attempt. Q Geometr hapter nswers

11 hapter nswers (continued) K : coring Guide Foundation drawing:. ample: F rthographic drawing: Front op ight tudent draws completel accurate figures. tudent draws generall accurate figures, but these ma contain a minor error. tudent draws figures containing significant errors. 0 tudent makes little or no attempt. umulative eview.. F.. G.. G.. J. 0. J.. G.. ample:,,,,,, c and,,,,,,, c he first is the natural numbers. In the second, the numbers,, and repeat.. Yes; the markings show the are congruent.. o; there are no markings.. Yes; ou can conclude the angles are supplementar from the diagram.. sometimes. never 0. ot acceptable; too general; ample: a ruler is a tool used for measuring lengths and is marked showing uniform units.. ample:. (-, -). (0, -). " 0 nswers Geometr hapter

12 hapter nswers ractice -. ample: It is :00 noon on a rain da.. ample: he car will not start because of a dead batter.. ample:. If ou are strong, then ou drink milk.. If a rectangle is a square, then it has four sides the same length.. If ou are tired, then ou did not sleep.. If =, then - = ; true.. If 0, then 0; true.. If m is positive, then m is positive; true. 0. If - =, then = ; true.. If 0, then point is in the first quadrant; false; (, -) is in the fourth quadrant.. If lines are parallel, then their slopes are equal; true.. If ou have a sibling, then ou are a twin; false; a brother or sister born on a different da.. siblings twin. Hpothesis: If ou like to shop; conclusion: Visit igeon Forge outlets in ennessee.. igeon Forge outlets are good for shopping.. If ou visit igeon Forge outlets, then ou like to shop.. It is not necessaril true. eople ma go to igeon Forge outlets because the people the are with want to go there.. rinking ustain makes ou train harder and run faster. 0. If ou drink ustain, then ou will train harder and run faster.. If ou train harder and run faster, then ou drink ustain. ractice -. wo angles have the same measure if and onl if the are congruent.. - = if and onl if =.. he converse, If n =, then n =, is not true.. figure has eight sides if and onl if it is an octagon.. If a whole number is a multiple of, then its last digit is either 0 or. If a whole number has a last digit of 0 or, then it is a multiple of.. If two lines are perpendicular, then the lines form four right angles. If two lines form four right angles, then the lines are perpendicular.. If ou live in eas, then ou live in the largest state in the contiguous United tates. If ou live in the largest state in the contiguous United tates, then ou live in eas.. ample: ther vehicles, such as trucks, fit this description.. ample: ther objects, such as spheres, are round. 0. ample: his is not specific enough; man numbers in a set could fit this description.. ample: aseball also fits this definition.. ample: leasing, smooth, and rigid all are too vague.. es. no. es ractice -. & and & are complementar.. Football practice is canceled for Monda.. F is a right triangle.. If ou liked the movie, then ou enjoed ourself.. If two lines are not parallel, then the intersect at a point.. If ou vacation at the beach, then ou like Florida.. not possible. amika lives in ebraska.. not possible 0. It is not freezing outside.. hannon lives in the smallest state in the United tates.. n hursda, the track team warms up b jogging miles. ractice -. U = M. m&qw = 0. = M. =. J. Given; ddition ropert of qualit; ivision ropert of qualit. Given; ddition; ubtraction ropert of qualit; Multiplication ropert of qualit; ivision ropert of qualit. ubstitution. ubstitution 0. ransitive ropert of qualit. mmetric ropert of ongruence. ransitive ropert of ongruence. efinition of omplementar ngles; 0, ubstitution;, implif;,, ubtraction ropert of qualit;, ivision ropert of qualit. Given; ( - ), ubstitution; -, istributive ropert; -, ubtraction ropert of qualit;, Multiplication ropert of qualit ractice m& =, m& =. m& =, m& =. m& =, m& = 0. m& = 0, m& = 0. m&m = ; m&mq = ; m&qm =. m& = m& = 0; m& = m& = ; m& = m& =. m&w = m&w, m&w + m&w = 0; m&w + m&w = 0; m&w = m&w eteaching -.. heck students work.. If ou hear thunder, then ou see lightning; statement: false; converse: false.. If our pants are jeans, then the are blue; statement: false; converse: false.. If ou are eating a tangerine, then ou are eating an orange fruit; statement: false; converse: true.. If a number is an integer, then it is a whole number; statement: true; converse: false.. If a triangle has one angle greater than 0, then it is an obtuse triangle; statement: true; converse: true.. If n =, then n = ; statement: true; converse: false. 0. If ou got an for the quarter, then ou got an on the first test; statement: false; converse: false.. If a figure has four sides, then it is a square; statement: true; converse: false.. If =, then " = ; statement: true; converse: true. eteaching -. If n = or n =-, then n =. If n =, then n = or n =-.. If two segments are congruent, then the have the same measure. If two segments have the same measure, then the are congruent.. If ou live in alifornia, then ou live in the most populated state in the United tates. If ou live in the most populated state in the United tates, then ou live in alifornia.. If an integer is a multiple of 0, then its last digit is 0. If an integer s last digit is 0, then it is a multiple of 0. Geometr hapter nswers

13 hapter nswers (continued). o; countereamples ma var. ample: giraffe is a large animal.. Yes; two planes intersect if and onl if the form a line.. Yes; a number is an even number if and onl if it ends in 0,,,, or.. Yes; a triangle is a threesided figure if and onl if the angle measures sum to 0. eteaching -. aw of etachment; the police officer will give arlene a ticket.. aw of llogism; if two planes do not have an points in common, then the are parallel.. aw of etachment; andon has a broken arm.. ot possible; a conclusion cannot be drawn from a conditional and its confirmed conclusion. (rad ma live in eoria, Illinois.). aw of etachment; the circumference is p. eteaching -. ddition ropert of qualit. mmetric ropert of qualit. istributive ropert. ubtraction ropert of qualit. ddition ropert of qualit. Multiplication ropert of qualit. ubstitution. ransitive ropert of qualit.,,,, eteaching -..,,,,,, or,,,,,,. ample: m&0 + m& = 0 b the ngle ddition ostulate. m& + m& + m& = 0 b the ngle ddition ostulate. m&0 + m& = m& + m& + m& b the ransitive ropert of qualit. ubtract from both sides, and ou get m&0 = m& + m&. ecause &0 is a right angle, m& + m& = 0. It is given that m& = m&. ubstitution, m& = 0. ivide both sides b, and m& =.. ample: ecause & and &0 form a straight angle, m& + m&0 = 0 b the ngle ddition ostulate. the definition of supplementar angles, & and &0 are supplementar. 0. ample: ecause &0 and & form a straight angle, m&0 + m& = 0 b the ngle ddition ostulate. &0 is a right angle, so 0 + m& = 0. ubtract 0 from both sides, and ou get m& = 0. o m& is a right angle b the definition of right angles. nrichment -. point. postulate. parallel. perpendicular. perimeter. plane. protractor. pentagon. ample: W, 0. X. ample: If WHJ stands for plane, then HXHWWW must be parallel because of the repetition of the letter W.. heck students work. nrichment -. If a kitchen is well-designed, then the distance in a straight line from the center point of the sink, refrigerator, or stove to another is between feet and feet. If the distance in a straight line from the center point of the sink, refrigerator, or stove to another is between feet and feet, then the kitchen is well-designed.. he statement can be written as a biconditional because both the conditional and its converse are true.. kitchen is considered to be well-designed if and onl if the distance in a straight line from the front center point of one of these three objects to another is between feet and feet. dditionall, the sum of the three triangle legs should be less than or equal to feet.. he definition is a good definition. he terms used are commonl understood. he definition is precise. he definition is reversible: If the distances in a straight line between the front center points of the sink, refrigerator, and stove are between feet and feet, and the sum of these distances is less than or equal to feet, then the kitchen is considered to be well-designed.. Yes; the distance from the sink to the refrigerator is. feet, the distance from the refrigerator to the stove is. feet, and the distance from the stove to the sink is feet. he sum of these distances is less than feet.. heck students work. nrichment -. p: one is a student at ichmond High chool; q: one takes geometr class; r: one studies logic.. If someone is a student at ichmond High chool, then he or she takes geometr class.. If someone takes geometr class, then he or she studies logic.. ll students at ichmond High chool stud logic.. ogic can be studied in classes other than geometr, such as algebra or trigonometr.. a; p: a figure is a square; q: a figure is a polgon; r: a figure is closed; if p q and q r then p r.. b; p: a figure is a square; q: a figure is a rectangle; r: a figure has four right angles; if p q and q r, then p r.. ll mathematicians enjo puzzles.. riangles have an angle sum of Mt. McKinle is 0,0 ft tall.. olan an recorded strikeouts over his career.. rapezoids are four-sided figures. nrichment - M U 0 I I nrichment -. he are complementar angles; if the eterior sides of two adjacent acute angles are perpendicular, then the angles are complementar.. &. he are equal; I U U I I V I I U I I I G I I I I V I V I U M I M I I I Y I U I U Q U I Y 0 nswers Geometr hapter

14 hapter nswers (continued) because & and & have equal measures, ou can use the definition of supplementar angles and the subtraction propert to show that the are equal bisects ; a segment that is perpendicular to another segment at its midpoint is the perpendicular bisector of the segment. hapter roject heck students work. heckpoint Quiz. hpothesis: + = 0; conclusion: =. hpothesis: if ou want to get good grades in school; conclusion: ou must stud hard. ample: It could be Ma.. ample: orn is a vegetable.. o; the lines must be in the same plane.. es. not possible. X, Y, and Z are coplanar.. If ou ran a good race, then our coach is happ. not efficient. 0. If the car is old, then it is heckpoint Quiz. =. &QM &. mmetric ropert. ubstitution ropert. ivision ropert. ddition ropert. & &, & &; Vertical ngles heorem. & &, & &; inear air and ransitive ropert of qualit. & &, & &; ubtraction ropert of qualit 0. & &, & &; Vertical ngles heorem hapter est, Form a. If a polgon has three sides, then it is a triangle. b. true a. If George lives in the United tates, then he lives in eas. b. false a. If two angles are congruent, then the are vertical angles. b. false. ddition ropert of qualit. ransitive ropert of qualit. ubtraction ropert of qualit. ivision ropert of qualit. efleive ropert of qualit a. 0 b. c. d. 0 e. f not possible. We win.. If the bus is late, then we will receive a tard penalt.. he sum of the measures of & and & is 0.. If a quadrilateral is a rectangle, then it has four right angles. If a quadrilateral has four right angles, then it is a rectangle.. rhombus has four congruent sides. 0. good definition. good definition. bat has wings. a. istributive ropert b. ddition ropert of qualit c. ivision ropert of qualit.... nswers ma var. ample: (, ). nswers ma var. ample: (, -) hapter est, Form a. If a polgon has five sides, then it is a pentagon. b. true a. If Mar lives in Minnesota, then she lives in Minneapolis. b. false. efleive ropert of qualit. ubstitution ropert of qualit. ivision ropert of qualit. istributive ropert a. b. c. d Martina is quick.. If I don t wear sunscreen while swimming, then I ll be in pain.. If a quadrilateral is a parallelogram, then it has two pairs of opposite sides parallel. If a quadrilateral has two pairs of opposite sides parallel, then it is a parallelogram.. rectangle has four right angles.. good definition. n octopus has eight legs. a. ivision ropert of qualit b. ubtraction ropert of qualit c. ivision ropert of qualit.. 0. nswers ma var. here are two possibilities. amples: (, -) or (-, ) lternative ssessment, Form K : coring Guide a. ample: If it is raining, then the lawn is wet. b. ample: If the temperature is below F, then the temperature is below freezing. tudent gives conditionals that meet the criteria and provides clear and accurate eplanations of wh the conditionals meet the criteria. tudent gives eamples and eplanations that are generall clear but ma contain errors. tudent makes significant errors in eamples or eplanations. 0 tudent makes little or no attempt. K : coring Guide ample: he definition cannot be written as a biconditional. school is a place where people are educated. tudent gives an eplanation and a definition that are accurate, clear, and complete. tudent gives an eplanation and a definition that are generall clear but ma contain errors. tudent makes significant errors in eplanation or definition. 0 tudent makes little or no attempt. K : coring Guide a. ample: If two angles have sides that are opposite ras, then the are vertical angles. &F and &G are angles whose sides are opposite ras. herefore, &F and &G are vertical angles. b. ample: &F and &G are vertical angles. ll vertical angles are congruent. herefore, &F &G. tudent gives accurate and complete eamples and eplanations of the aws of etachment and llogism. tudent gives eamples and eplanations that are generall correct but ma be unclear. tudent gives eamples or eplanations that contain major errors. 0 tudent makes little or no attempt. Geometr hapter nswers

15 hapter nswers (continued) K : coring Guide ample: V umulative eview. b. a.. G.. H.. F. 0. G.. J... If a polgon has three sides, then it is a triangle; true.. p m. ample: wo supplementar angles are not necessaril a linear pair.. * ). ubstitution ropert of qualit 0. (-, ). (-, ). ".. 0 m. 0. cm. ample: midpoint must be on the segment. n point that lies on the perpendicular bisector of a segment is equidistant from the endpoints. a. & and &V; &V and &V; &V and &; & and &; & and &V; & and &V b. & and &V; & and &V c. & is a right angle because the lines are perpendicular, so m& = 0. herefore, if ra Y bisects &, m&y =. tudent draws a diagram and gives responses that indicate a thorough understanding of the definitions involved. tudent shows logic in part c that is complete and accurate. tudent draws a diagram and gives responses that, while basicall accurate, contain some minor flaws, inaccuracies, or omissions. tudent draws a diagram or gives responses that contain significant inaccuracies or omissions. tudent displas flaws in logical argument in part c. 0 tudent makes little or no attempt. nswers Geometr hapter

16 hapter nswers ractice -. corresponding angles. alternate interior angles. same-side interior angles. alternate interior angles. same-side interior angles. corresponding angles. and, and, and, and. and, and. and, and 0. m = 00, alternate interior angles; m = 00, corresponding angles or vertical angles. m =, alternate interior angles; m =, vertical angles or corresponding angles. m =, corresponding angles; m =, vertical angles. = 0;, 0. = ;,. = 0;, a. lternate Interior ngles heorem b. Vertical angles are congruent. ongruence ractice - c. ransitive ropert of * ) a. same-side interior b. c. * ) Q d. same-side interior e. ame-ide Interior ngles f. * ) g. -. l and m, onverse of ame-ide Interior ngles heorem. none. and, onverse of ame-ide Interior ngles heorem. and HU, onverse of orresponding ngles ostulate. H and I, onverse of orresponding ngles ostulate. a and b, onverse of ame-ide Interior ngles heorem ractice -. rue. ver avenue will be parallel to Founders venue, and therefore ever avenue will be perpendicular to enter it oulevard, and therefore ever avenue will be perpendicular to an street that is parallel to enter it oulevard.. ot necessaril true. o information has been given about the spacing of the streets.. rue. he fact that one intersection is perpendicular, plus the fact that ever street belongs to one of two groups of parallel streets, is enough to guarantee that all intersections are perpendicular.. rue. pposite sides of each block must be of the same tpe (avenue or boulevard), and adjacent sides must be of opposite tpe.. ot necessaril true. If there are more than three avenues and more than three boulevards, there will be some blocks bordered b neither enter it oulevard nor Founders venue.. a e. a e. a e. a e 0. a e. a e. If the number of statements is even, then n. If it is odd, then n.. he proof can instead use alternate interior angles or alternate eterior angles (if the angles are congruent, the lines are parallel) or same-side interior or same-side eterior angles (if the angles are supplementar, the lines are parallel).. It is possible. l l ractice = ; = ; z =. a = ; b = ; c =. v = ; w = ; t = m = ; m = ; m =.... m = ; m =. isosceles. obtuse scalene. right scalene. obtuse isosceles 0. equiangular equilateral ractice -. = 0; = 0. n =. a = 0; b = F. F and.. 0 ractice -. = -. =- +. = -. =- -. = -. = -. = -. = l Geometr hapter nswers

17 hapter nswers (continued).... (, 0) (0, ) (0, ) (, 0) =- -. = + 0. = +. = +. = +. = -. =- -. =- +. = ; =-. = 0; =. =-; =-. =-; =.. (, 0) 0 (0, ). 0. (0, ) (, 0). (, 0) (0, ) (, 0) (0, ).. (, 0) 0 a. m = $0.0 b. the amount of mone the worker is paid for each bo loaded onto the truck c. b = $.0 d. the base amount the worker is paid per hour. = - + ractice - (0, ). neither;,? -. perpendicular;? - =-. parallel; - = -. parallel; - =-. perpendicular; = is a horizontal line, = 0 is a vertical line. parallel; - = -. neither;,? -. parallel; - = -. perpendicular; -? =- 0. neither; -,? -. neither; - -, -?- -. neither; -,? - -. neither;,? -. parallel; =. =. =- +. =- -. = +. = 0 0. = -. = ractice Q (0, ) (, 0) 0 nswers Geometr hapter

18 hapter nswers (continued)... ample: K b 0. ample: c. b. ample:.. a a b c b c b b b a eteaching - a. b. ample: c. d. ample: eteaching -. l m Given m = m = 0 ubstitution a If lines, then corresponding s are. and are supplementar. efinition of supplementar m + m = 0 ngle ddition ostulate Geometr hapter nswers

19 hapter nswers (continued). Given a b Vertical s are. ubstitution If corresponding s, then lines are parallel. eteaching -. raw a temporar line through that is perpendicular to, then draw a second, permanent line through that is perpendicular to the first. his line will be parallel to. epeat for. he two permanent lines will be parallel not onl to but to each other (heorem -).. raw two perpendicular lines through, then for each one draw a line perpendicular to it that runs through. he four lines will define a rectangle.. raw a line that runs through both and F, and then a second line at an acute or obtuse angle that runs through. raw a third line, parallel to the second and running through F, using the same procedure as in the ample and in ercise. In the same wa, draw a fourth line parallel to the first (at an desired distance from it). he four lines will define a parallelogram. eteaching -. : m = 0, m = 0; : m = 0, m = 0, m = 0; : m = 0, m = 0, m = 0. and are acute, equiangular, and equilateral; and are isosceles and obtuse;,,, and are right and scalene.. Q: m Q =, m Q = 0; Q: m Q = 0, m Q =, m Q = ; Q: m Q = 0, m Q = ; Q: m Q = 0, m Q =, m Q =. = = = = 0 mm; Q = Q = Q = Q = mm. Q, Q, Q, Q,,,, and are right and isosceles. eteaching -. and are interior angles; and are eterior angles.. m = ; m = 0; m = ; m = 0. is an interior angle; and are eterior angles; is neither.. m = 0; m = 0; m = 0; m = 0 eteaching - heck students graphs.. = -. =. =- -. = +. =- +. =. = =- +. = + 0. =. =- -. =-. = = - 0. = +. =- +. =- +. = eteaching - a. - b. a. b. - a. undefined b. 0 a. =- + b. = + c. a. =- - b. = - c. a. = + b. =- - c.. m JK =-; m M =-; parallel. m JK = ; m M = - ; perpendicular. m JK =-; m M =-; neither 0. m JK =-; m M = ; neither. m JK = ; m M = - ; perpendicular. m JK = ; m M = ; neither. m JK = ; m M =-; parallel. m JK undefined; m M = 0; perpendicular nswers Geometr hapter

20 hapter nswers (continued) eteaching -. ample:.. heck students work. Z X a b Y nrichment ) -. * is # ) * ) ) to..,,,,, or,,,,, ; is # to ; if two angles are congruent and supplementar, then each measures 0.. > ; aw of eflection. > ; lternate Interior ngles heorem. > ; aw of eflection. > ; ransitive ropert of ongruence nrichment -. = ample: ecause m =, m F = 0 - = ; l m because a pair of alternate interior angles are congruent.. and F; K. ample: and IGH are corresponding angles related to parallel segments and GH. K is the related transversal.. ample:,, and form a triangle, so 0 - ( + ) =, so m =. ecause and are congruent corresponding angles, b the onverse of the orresponding ngles ostulate. nrichment -. If the paper shifts during drawing, lines meant to be parallel will not be parallel, and lines meant to be perpendicular will not be perpendicular.. he rectangular shape gives the -square two perpendicular guide edges on which to line up.. heorem -0, which sas that if two lines are perpendicular to the same line, the are parallel to each other. line drawn with the -square is perpendicular to the edge on which the - square is lined up. wo lines drawn with the -square lined up on the same edge will be parallel to each other.. a d. If the -square is used to draw a line perpendicular to a vertical edge of the rectangular backing surface, and then is used to draw a line perpendicular to a horizontal edge of the backing surface, then the two lines will be perpendicular.. Figure would be eas; all the lines are parallel or perpendicular. Figures and would be hard to draw, since the lines are not all parallel or perpendicular to each other.. he will all be parallel to.. drafting machine can be used to draw slanted lines, not just vertical and horizontal ones, with precise control of the slant.. Figures and would be eas, because the lines have no more than two or three different slants. Figure would be harder, because lots of different slants are used to achieve the perspective effect. nrichment ngles have measures of 0, 0, or 0; M KF JG; MH K; H G; M K JF IG. nrichment umber of sides otal degree measure umber of diagonals nrichment -.. K nrichment -. (-, ). (-, ). (, ). (-, -). (-, -). (, -). (, 0). (, -). (-, ) 0. (, -). (-, -). (, ). (, -). (, 0). (0, ). (, ) -ais G 0 I... n (n ) H Y Y I M n(n ) -ais Geometr hapter nswers

21 hapter nswers (continued) nrichment -.,., and. pentagons heagons. scalene, acute triangle. o; refer to the answer to ercises,, and... isosceles, obtuse triangle hapter roject ctivit : aper Folding ; all triangles are right isosceles; es. ctivit : ploring triangle quadrilaterals F ctivit : nalzing ctivit : Modeling heckpoint Quiz. onverse of orresponding ngles ostulate. lternate Interior ngle heorem. ame-ide Interior ngles heorem. orresponding ngles ostulate. onverse of lternate Interior ngle heorem. Vertical ngles heorem. onverse of orresponding ngles ostulate. orresponding ngles ostulate. onverse of ame-ide Interior ngles heorem 0. = 0, = 0, z = heckpoint Quiz. pentagon; 0. heagon; = 0; =. quadrilateral; = ; = 0.. (0, ) (,0) (0,) (.,0) nswers Geometr hapter

22 hapter nswers (continued).. arallel; slopes are the same.. neither. erpendicular; the product of the slopes is neither hapter est, Form. true. true. false. false. true. false. true. true. nswers ma var. ample: m =, ame-ide Interior ngles heorem; m =, lternate Interior ngles heorem 0. nswers ma var. ample: m = 0, orresponding ngles ostulate then ngle ddition ostulate; m = 0, ame-ide Interior ngles heorem. nswers ma var. ample: m =, lternate Interior ngles heorem; m =, ame-ide Interior ngles heorem. nswers ma var. ample: m =, orresponding ngles ostulate; m = 0, ngle ddition ostulate. nswers ma var. ample: m =, orresponding ngles ostulate and ame-ide Interior ngles heorem; m =, orresponding ngles ostulate. nswers ma var. ample: m = 0, lternate Interior ngles heorem; m =, ame-ide Interior ngles heorem.. (,0) 0 (0, ) X. W and X. none. WZ and 0. none. WZ and. WZ and ; X and Y. = ; = 0. = 0; = 0; z = 0. = ; = ; z =. 0.. perpendicular. neither 0. parallel. = +. =- +. = + m hapter est, Form. true. true. false. true. false. m =, m =, alternate interior angles. m =, m =, same-side interior angles. m =, m =, alternate interior angles. 0.. and WZ. and WZ. and WZ. W and Z. =, =. w =, =, =, z =. 0.. neither 0. perpendicular. parallel. = -. = -. =- - lternative ssessment, Form K : coring Guide a. t a b b. ample: ; ; and are supplementar. c. m = ; m = 0, m =, m = 0, m =, m = 0, m = d. and are supp. Given and are supp. ngle dd. ost. upp. of the same are. a b If alt. int s are, then lines are. tudent draws an accurate diagram and supplies correct answers and a complete and accurate flow proof. tudent draws a figure or gives answers that contain minor errors. tudent draws a figure or gives answers that contain significant errors or omissions. 0 tudent makes little or no attempt. Geometr hapter nswers

23 hapter nswers (continued) K : coring Guide ample: K : coring Guide ample: Q Isosceles riangles quilateral riangles riangles tudent constructs an accurate figure. tudent constructs a figure that contains minor errors or omissions. tudent constructs a figure that contains significant errors or omissions. 0 tudent makes little or no attempt. K : coring Guide ample: a. For the figure given, = and = +. he lines are parallel because both lines have a slope of. c. For, the sum of the three angles is 0. Minor discrepancies are the result of measurement error and rounding error. d. For the figure given, is not regular. the distance formula, the sides are not congruent. tudent draws the figure accuratel, writes correct equations, and reasons logicall. tudent draws a figure, gives arguments, and writes equations that are mainl correct but ma contain minor errors. tudent presents work with significant errors. 0 tudent makes little or no attempt. tudent draws an accurate diagram. tudent draws a diagram that contains minor errors or omissions. tudent draws a diagram that contains significant errors or omissions. 0 tudent makes little or no attempt. umulative eview.. G.. F.. J.. F. 0. H.. J.. G.. G.. Given. ame-ide Interior ngles heorem 0. orresponding ngles ostulate. orresponding ngles ostulate. substitution. heck students work.. ketches ma var. he right angle must be between the equal sides.. o; a triangle cannot have two sides equal and no sides equal at the same time. nswers Geometr hapter

24 hapter nswers ractice -. m = 0; m = 0. m = 0; m =. m = 0; m = 0; m = 0; m = 0. J,, J. J,,. WZ JM, WX JK, XY K, ZY M. W J, X K, Y, Z M. Yes; GHJ IHJ b heorem - and b the efleive ropert of. herefore, GHJ IHJ b the definition of triangles.. o; Q V because vertical angles are congruent, and Q V b heorem -, but none of the sides are necessaril congruent. 0a. Given 0b. Vertical angles are. 0c. heorem - 0d. Given 0e. efinition of triangles ractice -. b. not possible. not possible. U XWV b. not possible. GHF b. MK KMJ b. Q b. not possible 0.. and. and. a. Given b. efleive ropert of ongruence c. ostulate. tatements easons. F FG, F FH. Given. F HFG. Vertical s are.. F HFG. ostulate ractice -. not possible. ostulate. heorem. heorem. not possible. not possible. ostulate. not possible. heorem 0. tatements easons. K M, K M. Given. JK M. Vertical s are.. JK M. ostulate. Q Given Q Given efleive ropert of. F. KHJ HKG or KJH HGK. M Q ractice - Q heorem. is a common side, so b, and b.. FH is a common side, so FH HFG b, and H FG b.. KJ M b, so K b.. Q is a common side, so Q Q b. Q Q b.. VX is a common side, so UVX WVX b, and U W b.. ZY and are vertical angles, so YZ b. Z b.. G is a common side, so G FG b, and FG G b.. JKH and KM are vertical angles, so HJK MK b, and JK K b.. is a common side, so Q b, and Q b. 0. First, show that and are vertical angles. hen, show b. ast, show b.. First, show FH as a common side. hen, show JFH GHF b. ast, show FG JH b. ractice -. = ; =. = 0; = 0. t = 0. r = ; s =. = ; = 0; z =. a = ; b = ; c = 0. =. a = 0; b = 0; c =. z = 0 0. ; F. G; G G. KJ; KIJ KJI. ;. ; J J. ; H H = 0; =. = 0; = 0. = ; = ractice -. tatements easons. #, # F. Given., are right s.. erpendicular lines form right s.. F,. Given. F. H heorem. tatements easons., are right s.. Given. Q. Given. Q Q. efleive ropert of. Q Q. H heorem. MJ and MJK are right s. erpendicular lines form right s. M MK Given MJ MJ efleive ropert of MJ MJK H heorem Geometr hapter nswers

25 hapter nswers (continued). GI JI Given HI HI efleive ropert of GHI JHI IHG IHJ H heorem GHI and JHI are right s. Given heorem - m GHI + m JHI = 0 ngle ddition ostulate. VW. none. m and m F = 0. GH JH. 0. UV or V U. m and m X = 0. m F and m = 0. GI # JH ractice -. ZWX YXW;. ;. JG FKH;. M;. F G;. UVY VUX;. common side:. F G G F common side: FG. common angle: M K H J 0. ample: tatements easons. X Y. Given. X #, Y #. Given. m X and. erpendicular lines m Y = 0 form right s... efleive ropert of. Y X. ostulate. ample: ecause FH G, HFG GF, and FG GF, then FG GFH b. hus, F GH b and H H, then GH FH b. eteaching -. b. c. a.. eteaching -.. heck students work.. ;. ; MQ. ; Q VU. : JMK MK. ; Q Q. ; YX WX eteaching -.. or. heck students work... JMK KM or JKM MK. UZ YZ. Y.. Y Y eteaching - ) a. QK Q; Q bisects KQ b. definition of bisector c. Q Q d. ostulate e.. tatements easons. M M,. Given. M M. efleive ropert of. M M. ostulate... tatements easons. bisects JH,. Given J H. J H. efinition of bisector.. efleive ropert of. J H. heorem. J H. eteaching -. ach angle is eteaching -. ample: =. cm, =. cm, Q =. cm, Q =. cm; not congruent. ample: =. cm, G =. cm, =. cm; G. ample: =. cm, =. cm, M =. cm; M. H heorem can be applied;.. H heorem cannot be applied.. H heorem can be applied; MU M.. H heorem can be applied; HF F or HF F.. H heorem can be applied; K H.. H heorem cannot be applied. nswers Geometr hapter

26 hapter nswers (continued) eteaching -. tatements easons. and Q. Given are right s; Q and. and Q. ight s are congruent... efleive ropert of. Q. heorem. Q. Vertical s are.. Q.. Q. heorem. ample: rove M Q b the heorem. hen use and vertical angles to show M Q b the heorem.. ample: rove b the ostulate. hen use and vertical angles to show F F b the heorem. nrichment - heck students work. amples shown nrichment - a. efinition of perpendicular lines b. KF G c. a. egment ddition ostulate b. + G = F + c. G d. lternate Interior ngles e. orresponding ngles f. JG g. nrichment -.. heck students work. a. b. he top angles are congruent because the fold bisected the right angles formed b the folds in steps and. he corners of the paper are right angles; therefore, those angles are congruent. he included sides are congruent because the fold in step found the midpoint of the width of the paper, thus creating two equal segments. a. b. he top angles are congruent because the fold bisected the right angles formed b the folds in steps and. he upper corners that became inside angles along the center line are right angles; therefore, those angles are congruent. he included sides are congruent because the fold in step found the midpoint of the width of the paper, thus creating two equal segments. a. he top angles are congruent because the fold bisected the right angles formed b the fold in step. he inside angles along the center line are right angles because the horizontal fold that formed them is perpendicular to the original fold in step. b. he included sides are congruent because the fold in step found the midpoint of the width of the paper, thus creating two equal segments. a. b. he top angles are congruent because the fold bisected the right angles formed b the fold in step. he inside angles along the center line are congruent because of the ngle ddition ostulate. he included sides are congruent because the fold in step found the midpoint of the width of the paper, thus creating two equal segments. nrichment -... ; ; ;. 0; 0; ;. efleive ropert of. heorem.. efinition of segments. efinition of segments 0. efinition of segments. ostulate.. 0 nrichment Geometr hapter nswers

27 hapter nswers (continued) nrichment - nrichment -. ample:. common angle:. ample:. common side: V I G H G U.. ample: GJ KH. K G H J. ample: HIM JIM. I I common side: IM H J M I M I H U 0... UV WV b ; V QV b ; VW VU b ; WQV UV b... heck students work.. heck students work. Y 0 M U M H F X I V I V hapter roject ctivit : Modeling triangle Yes; the brace makes two rigid triangles. ctivit : bserving heck students work. ctivit : Investigating tetrahedron ample: You could add in diagonals of the cube. heck students work. Finishing the roject heck students work. heckpoint Quiz.... not possible.. not possible. M Q, M Q,,, M Q,. lternate Interior ngles heorem. lternate Interior ngles heorem 0. heckpoint Quiz.,. Hpotenuse-eg heorem.. Q, Q, Q a. definition of a bisector b. efleive c. d. e. definition hapter est, Form. = 0; =. a = ; b = ; c =. H. not possible..... not possible 0.. not possible. heck students work; J, K Q,, JK Q, K Q, J... F. a. Given b. Given c. onverse of Isosceles riangle heorem d. ostulate e. f. Isosceles riangle heorem. ample: tatements easons. # ;is. Given midpoint of.. erpendicular lines form right s... efinition of midpoint.. efleive ropert of.. ostulate... ample: Given that X is the midpoint of and, X X and X X b the definition of midpoint. X X because all vertical angles are congruent. X X b the ostulate, and therefore b. nswers Geometr hapter

28 hapter nswers (continued) hapter est, Form. =, =. a = 0, b = 0, c = 0... H.. not possible. not possible. or 0. or.. ; ; F; ; F; F.. XY; YZ; XZ. X; Y; Z.. J a. Given b. Given c. ef. of equilateral d. efleive ropert of e. ostulate lternative ssessment, Form K : coring Guide ample: : M : : k kf F km km k k k k tudent s figures and information are clear and accurate. tudent s figures and information contain minor errors or omissions. tudent s figures and information contain significant errors or omissions. 0 tudent makes little or no attempt. K : coring Guide a. ample: cm K cm cm M cm b. ample: Using the thagorean theorem, show that K = M. hen K M b ostulate or ostulate. tudent s figures and eplanation are accurate and clear. tudent s figures and eplanation contain minor errors or omissions. tudent s figures and eplanation contain significant errors or omissions. 0 tudent makes little or no attempt. K : coring Guide ample: tatements easons.,. Given.. efleive ropert of.. heorem.... egment ddition ostulate tudent gives a proof that is accurate and complete. tudent gives a proof that contains minor errors or omissions. tudent gives a proof that contains significant errors or omissions. 0 tudent makes little or no attempt. K : coring Guide ample: he,, and ostulates are statements that are accepted as true without proof. he H and heorems, on the other hand, can be proved true, using postulates, definitions, and previousl proved theorems. tudent gives an eplanation that is thorough and correct. tudent gives an eplanation that is partiall correct. tudent gives an eplanation that lacks demonstrated understanding of the difference between a theorem and a postulate. 0 tudent makes little or no attempt. Geometr hapter nswers

29 hapter nswers (continued) umulative eview.. G.. H.. G.. G. 0. G.. J.. = 0; = 0. c, e, a, b, d or e, c, a, b, d. Given Given heorem efleive ropert. ample: he alarm sounds if and onl if there is smoke. If the alarm sounds, then there is smoke. If there is smoke, then the alarm sounds.., 0 nswers Geometr hapter

30 hapter nswers ractice - a. cm b. cm c. cm a.. in. b.. in. c.. in. a.. cm b.. cm c.. cm a. 0b.. GH, HI, GI. YZ, Q XZ, XY Q ractice -. WY is the perpendicular bisector of XZ right triangle.... equidistant 0. isosceles triangle.. )... right triangle. J is the bisector of J... ).. 0. ample: oint M lies on J.. right isosceles triangle ractice -. (-, ). (, 0). (, ). heck students work. he final result of the construction is shown.. altitude. median. none of these. perpendicular bisector. angle bisector 0. altitude a. (, 0) b. (-, -) a. ( 0, 0) b. (, -) a. (0, 0) b. (0, ) ractice -. I and III. I and II. he angle measure is not.. ina does not have her driver s license.. he figure does not have eight sides.. he restaurant is open on unda.. is congruent to XYZ.. m Y 0 a. If two triangles are not congruent, then their corresponding angles are not congruent; false. b. If corresponding angles are not congruent, then the triangles are not congruent; true. 0a. If ou do not live in oronto, then ou do not live in anada; false. 0b. If ou do not live in anada, then ou do not live in oronto; true.. ssume that m m.. ssume that UVW is not a trapezoid.. ssume that M does not intersect.. ssume that FGH is not equilateral.. ssume that it is not sunn outside.. ssume that is obtuse.. ssume that m 0. his means that m + m 0. his, in turn, means that the sum of the angles of eceeds 0, which contradicts the riangle ngle-um heorem. o the assumption that m 0 must be incorrect. herefore, m 0. ractice -. M,.,., Q.,.,.,. es; +, +, +. no; + 0. es; + 0. es; +, +, +. es; +, +, +. no; no; no; +. no;. +..,,.,,.,,.,, 0.,, J.,, eteaching - riangles will var. a. Q = b. M = c. YZ = a. =. b. = 0 c. UV = 0 a. Q = b. = c. UV = eteaching -. no. es. es. no.. eteaching -. ample:. ample: eteaching -. tep : ; tep : ; tep : F; tep : ; tep : ; tep :. tep : ssume F. tep : If F, then b the Isosceles riangle heorem, F. tep : ut F. tep : herefore, F. eteaching -. heck students work. he longest side will be opposite the largest angle. he shortest side will be opposite the smallest angle.. largest: F; smallest: F. largest: Q; smallest: Q. largest: ; smallest:. longest: F; shortest: F. longest: Q; shortest: Q. longest: V; shortest: Geometr hapter nswers

31 hapter nswers (continued) nrichment -.,. F. centroid. interior. It is not possible for the centroid to be on the eterior because all the medians are in the interior of the triangle.. F H. (0, ). (., 0). F(., ).. square units. he are equal.. :.,. Q 0. (., 0.). U(, -). V(., 0.). square units.. square units nrichment heck students work.. heck students work. Give hint: m =. nrichment -.. V 0 U. orthocenter. It is possible for the orthocenter to be in the interior, the eterior, or on the triangle itself. planations ma var.. Y I X 0. heck students work.. centroid nrichment -. If a fish is a salmon, then it swims upstream.. If a fish swims upstream, then it is a salmon.. If a fish is not a salmon, then it does not swim upstream.. If a fish does not swim upstream, then it is not a salmon.. ercises and. the salmon. the fish that swim upstream. U. U Q Q enter of Gravit 0.. U Q Z Q U 0 nswers Geometr hapter

32 hapter nswers (continued). If a quadrilateral is a rectangle, then the angles of the quadrilateral are congruent.. If the angles of a quadrilateral are not congruent, then the quadrilateral is not a rectangle.. If a quadrilateral is not a rectangle, then the angles of the quadrilateral are not congruent.. ll are true (,, and ).. the contrapositive nrichment -. U.. F.. U. F F.. U.. F. F. U.. he name of the mathematician is H (th a l ez). hapter roject ctivit : reating isoceles; median, angle bisector, perpendicular bisector ctivit : perimenting heck students work. ctivit : esigning heck students work. heckpoint Quiz. angle ) bisector, since M M. congruent, since QM is the angle bisector. 0, since M M.. a. b.. altitude. congruent heckpoint Quiz. Inv: If ou don t eat bananas, then ou will not be health. ontra.: If ou are not health, then ou don t eat bananas.. Inv.: If a figure is not a triangle, then it does not have three sides. ontra.: If a figure does not have three sides, then it is not a triangle.. uppose that there are two obtuse angles in a triangle. his would mean that the sum of two angles of a triangle would eceed 0. his contradicts the riangle-um heorem. herefore, there can be onl one obtuse angle in a triangle.. uppose that the temperature is below F and it is raining. ecause F is the freezing point of water, an precipitation will be in the form of sleet, snow, or freezing rain. his contradicts the original statement. herefore, the temperature must be above F for it to rain.. es; +, +. no;. +...,,., F, hapter est, Form a. If a triangle does not have three congruent sides, then it is not equiangular; true. b. If a triangle is not equiangular, then it does not have three congruent sides; true. a. If an isosceles triangle is not obtuse, then the verte angle is not obtuse; true. b. If the verte angle of an isosceles triangle is not obtuse, then the triangle is not obtuse; true. a. If two lines are not parallel, then the intersect; false. b. If two lines intersect, then the are not parallel; true.. =.. =. ample: F HG. I and III. II and III.,, 0.,,.,,.. ample: ssume that an equilateral triangle has a right angle. his means that one angle measures 0. his, in turn, means that each of the three angles is 0 because the triangle is equilateral and also equiangular. his means that the angle sum of the triangle is 0, contradicting the riangle ngle- um heorem. o the assumption that an equilateral triangle can have a right angle must be incorrect. herefore, an equilateral triangle cannot have a right angle.. ach leg must be greater than units in length.. ; 0. on. inside. outside. 0. X, X, X. (, ). (, -)... hapter est, Form ample: V =.,,.,,.. ; 0. inside. inside. on.., M, M, M,... lternative ssessment, Form K : coring Guide ample: ssume that Harold can create a triangular garden plot with sides of ft, ft, and ft. hen a triangle is possible with sides,, and. ut +, so b the riangle Inequalit heorem, no such triangle eists. herefore, the plan is impossible. tudent gives a clear and accurate eplanation. tudent gives an eplanation that contains minor inaccuracies. tudent gives an eplanation that contains significant flaws in logical reasoning. 0 tudent makes little or no attempt. K : coring Guide ample: he locus of points in a plane equidistant from a single point forms a circle, as shown in the first figure. he locus of points in a plane equidistant from the endpoints of a segment forms a perpendicular bisector of the segment, as shown in the second figure. Geometr hapter nswers

33 hapter nswers (continued) tudent gives an accurate eplanation and drawings. tudent gives a drawing or an eplanation that contains minor errors. tudent gives an eplanation or a drawing that contains significant errors. 0 tudent makes little or no attempt. K : coring Guide pressed as a conditional: If is an equilateral triangle with midpoints,, and F on sides,, and respectivel, then F forms an equilateral triangle. ample: F is an equilateral triangle. ach of F s sides is eactl the corresponding side of. If s sides are all the same length, then all of F s sides are all the same length as well. tudent accuratel states the conditional and the proof. tudent has minor errors in the statement of the conditional or in the proof. tudent gives a statement of conditional or a proof that contains significant errors. 0 tudent makes little or no attempt. K : coring Guide ample: is isosceles. = because is an isosceles triangle with =. he angle bisectors create two new angles at each verte of, each of which has a measure that of the angle bisected. hus m = m and m = m. ecause m = m, m = m, which makes an isosceles triangle. tudent gives accurate answers and justification. tudent gives answers containing minor errors. tudent gives a justification containing major logical gaps. 0 tudent makes little or no effort. umulative eview.. G.. H.. G.. H. 0. F.. J.. F... ample: he principal is giving a student an award.. uppose that an equilateral triangle has an obtuse angle. hen one angle in the triangle has a measure greater than 0. ut this contradicts the fact that in an equilateral triangle, the measure of each angle is 0. herefore, the equilateral triangle does not have an obtuse angle.. ample: 0. (-, ). Q 0.,, M nswers Geometr hapter

34 hapter nswers ractice -. parallelogram. rectangle. quadrilateral. parallelogram, quadrilateral. kite, quadrilateral. rectangle, parallelogram, quadrilateral. trapezoid, isosceles trapezoid, quadrilateral. square, rectangle, parallelogram, rhombus, quadrilateral. = ; = = = = 0. m = ; s = ; = M = ; = M =. f = ; g = ; FG = GH = HI = IF =. parallelogram. rectangle. kite. parallelogram ractice ; 0; 0. 0; 0; 0. ; ;. ; ; 0. ; 0;. ; ; 0; 0. ; ; ractice -. no. es. es. no. es. es. = ; =. = ; =. = ; = 0 0. = ; the figure is a because both pairs of opposite sides are congruent.. = 0; the figure is not a because one pair of opposite angles is not congruent.. = ; the figure is a because the congruent opposite sides are b the onverse of the lternate Interior ngles heorem.. Yes; the diagonals bisect each other.. o; the congruent opposite sides do not have to be.. o; the figure could be a trapezoid.. Yes; both pairs of opposite sides are congruent.. Yes; both pairs of opposite sides are b the converse of the lternate Interior ngles heorem.. o; onl one pair of opposite angles is congruent.. Yes; one pair of opposite sides is both congruent and. 0. o; onl one pair of opposite sides is congruent. ractice - a. rhombus b. ; ; ; a. rectangle b. ; ; ; a. rectangle b. ; ; 0; a. rhombus b. ; 0; 0; a. rectangle b. 0; 0; 0; 0 a. rhombus b. ; ; ; 0. Yes; the parallelogram is a rhombus.. ossible; opposite angles are congruent in a parallelogram.. Impossible; if the diagonals are perpendicular, then the parallelogram should be a rhombus, but the sides are not of equal length. 0. = ; HJ = ; IK =. = ; HJ = ; IK =. = ; HJ = ; IK =. =-; HJ = ; IK = a. 0; 0; ; b. cm a. 0; 0; 0; 0 b. in. a. ; 0; 0; b. 0 m. possible. Impossible; because opposite angles are congruent and supplementar, for the figure to be a parallelogram the must measure 0, the figure therefore must be a rectangle. ractice -. ;. ;. ;. ;. 0;. ;. =. = ; =. = 0. 0.; 0.. 0;. ;. 0; ;. 0; 0. 0; ;. =. =. = ; = ractice -. (.a, b); a. (.a, b); "a b. (0.a, 0); a. (0.a, b); "a b b b b b.. a 0. - a. a. - a. (a, b); I(a, 0). (a, b); M(a, -b); (-a, -b). (a, b); I(a, 0). (0, b); (a, b); (a, b). (-a, b). (-b, 0) ractice - "a b a. p q p b. = m + b; q = q (p) +b; b+q- q ; p p = +q- q q c. = r + p p d. = q (r + p) p + q - rp p ;= + p + q - rp q q q q ;= q + q; rp intersection at (r + p, q + q) e. r q f. (r, q) r r g. = m + b; q = (r) + b; b = q - q q ; r r = + q - q q r r h. = (r + p) + q - q q ; r = rp r + + q - rp q q q ;= q + q; intersection at rp (r + p, q + q) rp i. (r +p, q +q) a. (-a, 0) b. (-a, b) a b c. (-, ) b d. a a. (-a, 0) b. (-a, a) c. d. (a, -a) e.. he coordinates for are (0, b). he coordinates for are (a, 0). Given these coordinates, the lengths of and H can be determined: = "(a 0) (0 b) = ; H = "(0 a) (0 b) = "a b ; = H, so H. eteaching -.. amples:... parallelogram rectangle trapezoid p Geometr hapter nswers

35 hapter nswers (continued). rhombus. parallelogram. rectangle. isosceles trapezoid. rhombus. trapezoid 0. kite. rectangle. square eteaching -. tatements easons. arallelogram. Given.,. pposite sides of a parallelogram are congruent... efleive rop. of..... tatements easons. arallelogram ;. Given.. pposite angles of a parallelogram are... Isosceles riangle heorem.. ubstitution. tatements easons. arallelogram ;. Given.. pposite sides of a parallelogram are... ubstitution.. Isosceles riangle heorem. tatements easons. arallelogram ;. Given.. pposite angles of a parallelogram are... ubstitution.. If s of a are, sides opposite them are.. is isosceles.. ef. of isosceles triangle. tatements easons. Isosceles trap. ;. Given. ;. efinition of isosceles trapezoid.. ransitive ropert.. lt. int. s are... ransitive ropert. is isosceles.. efinition of isosceles triangle.. efinition of isosceles triangle.. ransitive ropert eteaching -. no. no. no. es. tatements easons.,,. Given.. ubstitution. is a. If one pair of opposite parallelogram. sides is both congruent and parallel, then the quadrilateral is a parallelogram.. tatements easons.,. Given,.. If s of a are, sides opposite them are... ubstitution. is a. If both pairs of parallelogram. opposite sides are, then the quad. is a parallelogram. eteaching -. m = 0; m = 0; m = 0. m = 0; m = 0; m = 0; m = 00. m = 0; m = 00; m = 0; m = 0. m = 0; m = 0; m = 0. m = ; m = ; m = ; m = 0. m = ; m = eteaching ample: tatements easons. M. Given. M. heorem -.. efleive rop. of. M. ostulate. M.. M M. efleive rop. of. M M. ostulate. Q MQ.. Q MQ. Vertical angles are. 0. Q MQ 0. heorem ample: oth MQ and Q have the same angle measures, but their sides have different lengths. eteaching -. Q( + k, m). X(-a, 0); W(0, -b). (a, -a); (0, -a). ach side has length a ", so it is a rhombus. ne pair of opposite sides has slope of, and the other pair has slope of -. herefore, because ()(-) =-, the rhombus has four right angles and is a square.. ach side nswers Geometr hapter

36 hapter nswers (continued) has length of "a a. herefore, the figure is a rhombus.. ( - k, m) eteaching -. ach diagonal has length "c (b a).. he b c b a c midpoints are (, ) and (, ). he line connecting the midpoints has slope of 0 and is therefore parallel to the third a b a b side.. he midpoints are (, 0), (a, ), (, b) and (0, ). he segments joining the midpoints each have length a b a b "a b.. he midpoints are (, ), (-, ), a b a b (-, - ), and (, - ). he quadrilateral formed b these points has sides with slopes of 0, 0, undefined, and undefined. herefore, the sides are vertical and horizontal, and consecutive sides are perpendicular.. he median meets the base at (0, 0), the midpoint of the base. herefore, the median has undefined slope; i.e., it is vertical. ecause the base is a horizontal segment, the median is perpendicular to the base. a a d e b d c e. he midpoints are (, 0), (, ), (, ), b c e and (, ). ne pair of opposite sides has slope of d, c and the other pair of opposite sides has slope of b a. herefore, the figure is a parallelogram because opposite sides are parallel. nrichment -. ome. o. ome. ll. o. ome. o. ome. ll 0. ome. ome. o. o. ome. ll. ll. ome. o. ome 0. ll nrichment -., F, HG, FIH, IG, FH., F, HG, FIH, IG, FH, F, FIG, HG, IH., F, HG, FIH, IG, FH, F, FIG, HG, IH, H, G, FI, FH, F,, FHG, IH. HG, IH,, F, H, I, HFI, GFH, G, GFH, IH, FI. pentagon, scalene triangle, two rectangles, two trapezoids, two isosceles right triangles. F nrichment - a. Given b. efinition of a regular heagon c. d. e. Given an two distinct points, there is a unique line segment with these points as endpoints. f. efinition of a diagonal g. efinition of a regular heagon h. efleive ropert of ongruence i. j. k. If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. l. efinition of a regular heagon m. n. o. efinition of a regular heagon p. q. r. If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. s. efinition of a parallelogram. parallelogram can be constructed in an octagon b drawing the diagonals as shown. ther answers are possible. nrichment -.. cm; JK is a square, so = J =. cm. ecause J = - J, J =. -. =. cm.. ; M is a rhombus, so M #. o m M = 0 and m M = ; therefore m M =... cm; because m M = = m M, J. o J is a parallelogram, and J = =. cm... cm; because m M = = m M, M. M therefore is a parallelogram, so M = =. cm.. ; because m M = m M, m M =.. 0; because JK is a square, m JK = 0. ecause m JK + m JM = 0, m JM = 0.. ; m M + m M = m M =.. o; the base angles of the triangle are not congruent.. parallelogram 0. rhombus. square. rectangle nrichment -. square units. square units. When ou actuall cut out the shapes and reassemble them, ou find that the diagonal of the rectangle is not a straight line. rapezoid IV does not quite meet the top edge of triangle I, and similarl there is a space between trapezoid III and triangle II. he etra space represents the etra one square unit of area.. he areas differ b square inches. 0 in. 0 in. in. in. in. 0 in. 0 in. in. Geometr hapter nswers

37 hapter nswers (continued) nrichment (; 0 ). (; 00 ). (.; ). (; 0 ). (; ). (; 0 ). (; 0 ). (; ). (; 0 ) nrichment -. (0, ). (0, ). (, ). (, ). he -coordinate of each point decreased b.. (, 0). (-, -). (-., -.). dd to the -coordinate. 0. he -coordinate of each point decreased b.. he -coordinate decreased b.. (, ). M(0, ), (, ). rectangle. ither all the -coordinates or all the -coordinates change b a constant amount. hapter roject ctivit : oing heck students work. ctivit : nalzing. he effective area is rectangular.. he effective area is rectangular. he effective area is larger in Figure than in Figure because the diagonal is longer than the face.. You should tie the string to a vertical stick of the kite.. If the faces of the kite were unchanged, one diagonal of the rhombus is longer than the diagonals of the square, so the effective area would increase. ctivit : esearching heck students work. heckpoint Quiz. 0, 0, 0.,,.,,. square. = 0, =. =, =. 0. heckpoint Quiz I H 0 G F =, =. =, =. =, =. rue; the are the onl quadrilaterals that possess these properties.. False; onl two triangles at a time are congruent.. (n +, m). (k, 0) hapter est, Form.... trapezoid rhombus rectangle kite. cm. in.. m. 0. = ; = = 0; = 0 a b c.. (a, 0); (b, c); (, ). (-c, 0); c b a b (0, b); (-, ). (0, b); (a, 0); (, ). ;. 0;. 0; 0. ; 0. ;. ;. he lengths of segments,, and are: = "j k,= "k l,= l + j. hus, nswers Geometr hapter

38 hapter nswers (continued) the perimeter of is l + j + "j k + "k l. j k he midpoints of segments,, and are: M(-, ), l k l j (, ), (,0). he lengths of segments M,, and M are: M = (l + j), =, M = "j k. hus, the perimeter of M is "k l (l + j + "j k + "k l ), which is half the perimeter of.. parallelogram. kite. rectangle. parallelogram. square. rhombus 0. isosceles trapezoid hapter est, Form. square. parallelogram. kite 0 0. trapezoid. in.. cm. m. =.. = ; = 0. =. ; 0. 0;. ;. 0; ;. 0; ;. ; ;. parallelogram. rhombus. trapezoid 0. square lternative ssessment, Form K : coring Guide amples: a.,,,,,,,, =, =,,, b.,, F tudent lists all statements accuratel in part a and gives the correct answers in part b. tudent gives mostl correct answers but with some errors. tudent gives answers that fail to demonstrate understanding of the properties of parallelograms. 0 tudent makes little or no effort. K : coring Guide arallelograms ectangles quares hombuses tudent gives accurate and complete answers and diagram. tudent gives answers and a diagram that are mostl accurate. tudent gives answers or a diagram containing significant errors. 0 tudent makes little or no effort. K : coring Guide = 0 (iagonals of a kite are #.) = (ef. of isos. trapezoid) z = (ase angles of isos. trap. are.) tudent gives correct answers and reasons. tudent gives mostl correct answers and reasons. tudent gives mostl incorrect answers and reasons. 0 tudent makes little or no effort. K : coring Guide a. Q = ( - a, ); = (, - a) b. lope of 0 a = 0 =. lope of Q = ( a) = -. ecause the product of their slopes = -, # Q. tudent gives correct coordinates and a valid proof. tudent gives answers or a proof that contains minor errors. tudent gives incorrect coordinates in part a or a poorl constructed proof in part b. 0 tudent makes little or no effort. Geometr hapter nswers

39 hapter nswers (continued) umulative eview.. J.. H.. J.. G. 0. G.. J.. 0., 0,. roof: b the definition of a rhombus. lso,. herefore, b the heorem... Q 0. ample: he construction of the undercarriage of a bridge; it is a combination of triangles, which are the strongest geometric polgon. M 0 nswers Geometr hapter

40 hapter nswers ractice -. :. ft b 0 ft. ft b ft. ft b ft. ft b ft. ft b 0 ft. true. false. true 0. true. false. true. true. false. false :. :. :.. ractice -. XYZ, with similarit ratio :. QM, with similarit ratio :. ot similar; corresponding sides are not proportional.. ot similar; corresponding angles are not congruent.. KM, with similarit ratio :. ot similar; corresponding sides are not proportional.. I.. J ft.. in... cm. 0 m ractice -. X XQ because vertical angles are. (Given). herefore X XQ b the M X XM ostulate.. ecause W = W = =, MX W b the heorem.. QM M because vertical s are. hen, because QM M M = M =, QM M b the heorem.. M (Given). ecause there are 0 in a triangle, m J = 0, and J. o MJ b the ostulate.. ecause X X X = X and X = X, X = X. X X because vertical angles are. herefore X X b the heorem.. ecause = = and XY = YZ = ZX, XY = YZ = ZX. hen XYZ b the heorem ft ractice -... ". ". 0". ". h.. 0. a. b. c. = ; = ". = "; = ".. = "; = ". = "; = "; z = ". = ; = ";z = ". " in. ractice - JG.. H.. J. JI = ; =.. = ; =. = ; =... 0 eteaching -. apples. gallons. emploees. about 00 picture frames ,0. 00 eteaching -. KM Q; :. XYZ; 0 :. not similar. not similar. M QIK; :. not similar. similar with ratio 0 :. not similar. similar with ratio : eteaching -. ZYX b. not similar. QU I b. b. not similar. XQ b. es; b or b. o; the verte angles ma differ in measure.. ll congruent triangles are similar with ratio :. imilar triangles are not necessaril congruent. eteaching = ; = ; z =. = ; = ; 0 z =. =.; =. = ; =. = "; = "; z = ". = "0; 0 = "; z = ". eteaching -. =.. =. =.; =.. = ; =. =. = ; =. = ; =. Y = nrichment -. 0 lb. $0; costs have not changed during the past ear.. $0,000. $,0. $. $0. 0 d nrichment -.. heck students work. Geometr hapter nswers

41 hapter nswers (continued) nrichment -. tatements easons. is a trapezoid. Given.. ef. of a trapezoid.. Vertical angles are... lt. int. angles are... imilarit ost.. tatements easons., U, and V are mdpts. Given. U V, UV,. riangle Midsegment and V U heorem. UV and UV are. ef. of parallelogram parallelograms.. UV. pposite s of a parallelogram are.. UV. pposite s of a parallelogram are.. Q VU. imilarit ost.. Using the distance formula, = "0 or ", = " or ", =, = 0, = "0 or ", and = "0 or ". If the two triangles are similar, then corresponding sides are proportional. herefore, " " = 0 = = = = = = =. " " ecause = =, b the imilarit ostulate. nrichment -. he triangles have two congruent angles; imilarit ( ) ostulate.. =. =. 00 d. 00 ft. ft. m nrichment HIM F YU hapter roject ctivit : oing heck students work. ctivit : nalzing tage : ; tage : ; tage : ; ; ctivit : oing heck students work. ctivit : hinking he entire snowflake at each stage will be inside the regular heagon. he area of the heagon is square units, so the area of the snowflake is alwas less than square units. ctivit : Modeling heck students work. heckpoint Quiz.. :.... M.. ft. no; 0. 0 mi heckpoint Quiz. es; similarit postulate. es; similarit postulate.... =., =.... hapter est, Form..... XYZ b the ostulate. not similar. KM Q b the heorem. 0 ft. 0. " heck students work.. If two parallel lines are cut b a transversal, alternate interior angles are congruent. herefore, b the ostulate. 0. = ; = ";z = ". = ; = hapter est, Form Q b heorem. M FWY b ostulate. G K b heorem. not similar.. m :. :. =. = 0; =... lternative ssessment, Form K : coring Guide imilarit ostulate and and imilarit heorems; check students work. tudent gives correct answers and accurate drawings and eplanations. tudent gives mostl correct answers, drawings, and eplanations. tudent gives answers, drawings, and eplanations that contain significant errors. 0 tudent makes little or no effort. 0 nswers Geometr hapter

42 hapter nswers (continued) K : coring Guide laim is true, and laim is false. laim is partl true; the triangles are similar, but the ratio of the areas is :, not :. tudent gives correct answers and provides logical reasons to support the answers. tudent gives mostl correct answers, but the work ma contain minor errors or omissions. tudent gives answers that contain significant errors. 0 tudent makes little or no effort. ecause the ratio of heights is not close to the ratio of head diameter, the bab and the adult are not similar. (he bab s head will grow more slowl than the rest of the bod.) tudent gives a mathematicall sound analsis. tudent gives an eplanation that contains minor errors or omissions. tudent gives an eplanation containing significant errors or omissions. 0 tudent makes little or no effort. K : coring Guide ne method is to pick an point on land and measure and. hen select a fraction (sa, ), and mark X and Y halfwa from to and, respectivel. hen measure XY. hen =? XY. tudent gives a mathematicall correct procedure and eplanation. tudent gives a procedure and eplanation that ma contain minor errors. tudent gives an invalid procedure or a valid procedure with little or no eplanation. 0 tudent makes little or no effort. K : coring Guide omparing the adult to the bab, we have: ratio of heights. : ratio of head diameter. : X? Y umulative eview.. G.. G.. H.. F. 0. J.. H.. F. 0. ongruent triangles have congruent corresponding angles and congruent corresponding sides, so the have a similarit ratio of :. imilar triangles have congruent corresponding angles but do not necessaril have congruent corresponding sides.. ample:. If cows do not have si legs, then pigs cannot fl. Geometr hapter nswers

43 hapter nswers ractice -. ". ". ". ". ". ". in.. ft. cm 0. m. acute. right. obtuse. right. obtuse. acute ractice -. = ; = ". a =.; b =." 0" ". c = ; d =. ". " "... = ; = ". " 0. cm.. in.. 0. ft, ft. 0 in. 0" ". w = ; = ; = ";z =. a = ; b =. p = ";q = ";r = ; s = " ractice -. tan = ; tan F =. tan = ; tan F =. tan = ; tan F = ractice - "0. sin = ; cos =. sin = ; cos = ". sin = ; cos =. sin = ; cos = " ". sin = ; cos =. sin = ; cos = ractice - a. angle of depression from the birds to the ship b. angle of elevation from the ship to the birds c. angle of depression from the ship to the submarine d. angle of elevation from the submarine to the ship a. angle of depression from the plane to the person b. angle of elevation from the person to the plane c. angle of depression from the person to the sailboat d. angle of elevation from the sailboat to the person.. ft.. ft.. ft.. d.. m.. m a. ractice -..0,.0. -., -..., mi/h;.º north of east.. m;.0º south of west.. m/sec;.º south of east. º north of east. 0º west of south. º west of north 0a., 0b. a., - b. a. -, - b.. ample: W. ample: W 0 0 ft b. ft Geometr hapter nswers

44 hapter nswers (continued) eteaching -.. ample:. ample:,, and. he are equal.. he sum of the squares of the two legs equals the square of the hpotenuse.. ". ".. eteaching -.. ample:. ample:. and. ample:. and. ample: he are approimatel equal.. heck students work.. d = ";e = " ". f = ; g =. = "; = 0. u = ";v = eteaching m =.; m =.. m =.; m =.. m = 0.; m =. 0. m =.; m = eteaching -. F =., F =.. XZ =., YZ = 0. =., =.... 0; 0..;...;...;. a.. b. " eteaching -.. ft.. ft....0 ft.. ft.. ft eteaching -.. mi/h at. off directl across a. 0 mi b.. west of due north. he bird s path shifts.0 east of due south.. mi/h nrichment - Friar uck is the winner. using the thagorean heorem, Friar uck s string length is " + " or 0. in. ittle John s string length is " + " or. in. nrichment -. ". ". " or. ". or or 0 nrichment ; definition of supplementar angles. ; vertical angles are congruent.. ; vertical angles are congruent... -,. 0, 0 0. o; because the slopes of the lines are opposite reciprocals, the lines must be perpendicular.. 0., 0. nrichment -.. cm.. cm.. cm he are equal.. he sides of a triangle are proportional to the sides of their opposite angles km.. km. ighthouse Q; it is closer. nswers Geometr hapter

45 hapter nswers (continued) nrichment -. the top of the tower. M. 0 M 00 M. the angle of elevation from to the top of the tower; ; scalene right triangle. the angle of elevation from to the top of the tower; - -0 ; isosceles right triangle. tan 0 =. tan = 00. = tan 0(); = tan ( + 00).. 0..; the height of the tower. es nrichment -. (0., 0.). (-0.,.0). (-.,.0).. km;. south of east hapter roject ctivit : uilding and are complementar, and and are complementar, so m = m. line of sight string horizon line edge of protractor ctivit : Measuring heck students work. ctivit : omparing heck students work. ctivit : esearching heck students work. heckpoint Quiz. obtuse. right. acute. tan =, sin =, cos = ; tan =, sin =, cos =. tan =, sin = 0., cos = 0.; tan = 0., sin = 0., cos = 0.. tan = 0., sin = 0., cos = 0.; tan =., sin = 0., cos = heckpoint Quiz. (, ). (, -). (-0, )..,. north of east..,. south of west. 0.,. north of west.. ft hapter est, Form. right. acute. obtuse. Í.. Í; Í. ; Í " ". sin = ; cos = ; tan = " ". sin = ; cos = ; tan = a. angle of elevation from the person to the top of the tree b. angle of depression from the top of the tree to the person c. angle of elevation from the top of the tree to the birds X d. angle of depression from the birds to the top of the tree. 0 ft. ft. ample: he side opposite X is the same as the side adjacent to Y. he side adjacent to X is the same as the side opposite Y. Z Y opposite /X adjacent /Y o, tan = adjacent /X = opposite /Y = tan...,.. 0., mi; south of west., 0.,. ample: hapter est, Form. right. obtuse. Í. Í; Í " ". sin = ; cos = ; tan = " ". sin = ; cos = ; tan = a. angle of depression from the cloud to the person in the castle b. angle of elevation from the person in the castle to the cloud c. angle of depression from the person in the castle to the person on the ground d. angle of elevation from the person on the ground to the castle. 0 ft opposite adjacent. sin = hpotenuse and cos = hpotenuse sin opposite adjacent X cos = hpotenuse hpotenuse = = tan. -.,...0 mi;. north of east., 0 opposite hpotenuse hpotenuse? adjacent = opposite adjacent Z Y Geometr hapter nswers

46 hapter nswers (continued). ample: K coring Guide: a. W W W, - ;, b. W W W a b a, ; b, c. a south of east; b north of east tudent gives correct answers. tudent gives answers that are mostl correct. tudent gives mostl incorrect answers. 0 tudent makes little or no effort. lternative ssessment, Form K coring Guide: =. tudent gives correct answer and shows a valid method. tudent gives generall correct work that contains minor errors. tudent gives an incorrect answer, and the method is not correct. 0 tudent makes little or no effort. K coring Guide: ft tudent gives a correct answer and shows a valid method. tudent gives generall correct work that contains minor errors. tudent gives an incorrect answer, and the method is not correct. 0 tudent makes little or no effort. K coring Guide: =. mi; = 0. mi tudent gives correct answer and shows a valid method. tudent gives generall correct work that contains minor errors. tudent gives an incorrect answer, and the method is not correct. 0 tudent makes little or no effort. umulative eview.. H.. G.. G.. J. 0. F.. J.. 0. ample: You ma have used the branches to climb into the tree.. ample: uppose that an equilateral triangle has an obtuse angle. hen one angle in the triangle has measure greater than 0. his contradicts the fact that, in an equilateral triangle, the measure of each angle is 0.. ample: he side opposite the 0 angle is also the hpotenuse, and there are two adjacent sides.. ample: he sum of the three angles of a triangle is 0. n a sphere, the sum of the three angles will be greater than 0. nswers Geometr hapter

47 hapter nswers ractice -. o; the triangles are not the same size.. Yes; the heagons are the same shape and size.. Yes; the ovals are the same shape and size. a. and F b. and, and, F and F, F and F a. M and b. M and M, and, M and M. (, ) ( -, - ). (, ) ( -, - ). (, ) ( -, - ). (, ) ( +, - ) 0. (, ) ( -, + ). (, ) ( +, + ). W (-, ), X (-, ), Y (, ), Z (, ). J (-, 0), K (-, ), (-, -). M (, -), (, -), (, -), Q (, -). (, ) ( +., +.). (, ) ( +, - ). (, ) (,). (, ) ( +, + ) a. (-, -) b. (0, ), (-, ), Q (, ) ractice -. (-, -). (-, -). (-, -). (, -). (, -). (, -) a. b. a. b. J I K K J X Z W I Y X W (-, ). (-, 0). (0, -) ractice - 0,. I. I. I. GH. G.. U Y Z U Geometr hapter nswers

48 hapter nswers (continued).. ractice -. he helmet has reflectional smmetr.. he teapot has reflectional smmetr.. he hat has both rotational and reflectional smmetr.. he hairbrush has reflectional smmetr.. Q Q ' ' Q M Q' M Q Q Q Q' ' ' Q M M Q M Q. his figure has no lines of smmetr line smmetr X X line smmetr and rotational smmetr line smmetr and 0 rotational smmetr line smmetr and rotational smmetr 0 rotational smmetr line smmetr nswers Geometr hapter

49 hapter nswers (continued)... line smmetr and rotational smmetr... line smmetr rotational smmetr ractice -. (-, -), M (-, 0), (, -), (0, -). (-0, -0), M (-, 0), (0, -), (0, -). (-, -), M (-, 0), (, -), (0, -).... es. no. no 0.. K H X. (-, -), Q (-, 0), (0, -). (-, ), Q (,- ), (,). (-, ), Q (, ), (-, ). (-, ), Q (-, 0), (0, ) ractice -. I. II. III. IV.. I. II. III. IV..... J J m m m J m m Geometr hapter nswers

50 hapter nswers (continued) reflection. rotation. glide reflection. translation ractice translational smmetr line smmetr, rotational smmetr, translational smmetr, glide reflectional smmetr translational smmetr line smmetr, rotational smmetr, translational smmetr, glide reflectional smmetr line smmetr, rotational smmetr, translational smmetr, glide reflectional smmetr rotational smmetr, translational smmetr nswers Geometr hapter

51 hapter nswers (continued).. amples:. 0.. Y Z X. X Z. es. es. no. es. no. no eteaching -.. heck students work.. (0, -), (, ), (, -), (, -). (, -), (, ), (, 0), (, -). (, ), (, ), (, ), (, ) eteaching -.. heck students work.. reflection across -ais: F (-, -), G (-, -), H (-, -); reflection across -ais: F (, ), G (, ), H (, ). reflection across -ais: (, -), (, -), (, -); reflection across -ais: (-, ), (-, ), (-, ). reflection across -ais: J (-, ), K (-, ), (-, ); reflection across -ais: J (, -), K (, -), (, -) eteaching -... Y Y. ample: Y Z Z X X Z X X Y Y Z Z eteaching -. two lines of smmetr (vertical and horizontal), 0 rotational smmetr (point smmetr). one line of smmetr (horizontal). one line of smmetr (vertical). two lines of smmetr (vertical and horizontal), 0 rotational smmetr (point smmetr). one line of smmetr (vertical). one line of smmetr (vertical) eteaching -. heck students work.... Y X Z X Geometr hapter nswers

52 hapter nswers (continued). eteaching -. translation. reflection. rotation. glide reflection. rotation. glide reflection. reflection. translation eteaching -.. ample: line smmetr across the dashed lines, rotational smmetr around points, translational smmetr, glide reflectional smmetr line smmetr across the dashed lines, rotational smmetr around points, translational smmetr, glide reflectional smmetr. line smmetr across the dashed lines, rotational smmetr around points, translational smmetr, glide reflectional smmetr nrichment -. (, ) ( -, - ). (, ) ( +, - ). (, ) ( - 0, + ). (, ) ( -, - ). Foster. es. Wilson. (, ) ( -, - ). nrichment -. (-, -). (-, 0). (, ). the midpoint formula: M = (, ).. = +. =- ' 0 Q' Q K J 0 J' K' ' ' ' rotational smmetr around points, translational smmetr 0 nswers Geometr hapter

53 hapter nswers (continued). =-. = 0. =- -. V' = + M 0 ' Q' ' M' 0 ' 0 F U' W' ' 0 F' ' ' ' Q ' U ' V W. =- - nrichment -. (0, -, ). (-, -, ). (-, -, 0). (0, -, 0). (0, -, 0). (0, -, ). (-, -, ). (-, -, 0). (, 0, ) 0. (, -, ). (, -, 0). (, 0, 0). (, 0, 0). (, 0, ). (, -, ). (, -, 0) nrichment - G' H' 0 H G I K' J'. es; rotational and point smmetr. no. es; rotational and point smmetr. es; rotational and point smmetr. no. no. diamonds.. even of diamonds; this does not have smmetr because the diamond in the middle is toward either the bottom or top of the card, and when ou rotate the card 0, the position will be reversed. 0. ll the face cards have smmetr.. es;,, 0. o; ample: When ou look at the card one wa, three of the points of the hearts are pointing down, and five are pointing up. When ou rotate the card 0º, five of the points of the hearts are pointing down, and three are pointing up.. o; because the number and suit of each card are placed in opposite corners, none of the cards have line smmetr.. You can add a backward with the small club below it to the two empt corners to create line smmetr, or ou can remove the with the small club below it from each of the two corners. nrichment - a. (, 0, ) b. (0, 0, ) c. (0,, ) d. (,, ) e. (, 0, 0) f. (0, 0, 0) g. (0,, 0) h. (,, 0) a. (, 0, ) b. (0, 0, ) c. (0,, ) d. (,, ) e. (, 0, 0) f. (0, 0, 0) g. (0,, 0) h. (,, 0). (0, 0, 0).. ach edge in the image is double that in the preimage.. amples: hree faces are coplanar for image and preimage; three faces are parallel; each face in the image has two times the perimeter and four times the area of the corresponding face in the preimage.. urface area of image = sq. units; surface area of preimage = sq. units; the ratio of the surface areas is :. I' J K Geometr hapter nswers

54 hapter nswers (continued). Volume of image = cubic units; volume of preimage = cubic units; the ratio of the volumes is :.. ilations in three dimensions are proportional to dilations in two dimensions. nrichment -.. heck students work. 0. -; rotation; twice nrichment - I M G I I F hapter roject ctivit : Investigating a. vertical b. vertical c. both ctivit : Modeling a. ncient gptian ornament b. riental design c. Greek vase design G I F I F I M M I I I I G I M G I I M 0 Y M M F I I I Y ctivit : Investigating a, b ctivit : lassifing a. mg b. ctivit : reating heck students work. heckpoint Quiz. o; the figures are not the same size.. es. es. (, ) ( -, - ). units right and units down. a resultant translation of units right and units up... ', ' ' ' 0. (, -), Q (, -) and (, 0) heckpoint Quiz. line smmetr; rotational; 0. line smmetr; rotational; 0. line smmetr ' H. rotational; 0. translational; no turns. rotational; preimage turns. translational. rotational, reflectional, translational. translational hapter est, Form. (, 0), Q (0, -), (, -), (, -). (-, -), Q (, -), (, -), (-, -). (0, -), Q (, -), (, -), (, -). (-., 0), Q (-, -), (-., -), (-., -). (, ), Q (, 0), (, -), (, -). (-, ), Q (-, ), (-, -), (-, -). glide reflection. translation. translation 0. rotation ' ' ' ' nswers Geometr hapter

55 hapter nswers (continued).. '. line smmetr. rotational smmetr a. n two of the following: HIKMUVWXY b. n two of the following: HIXZ. ranslations can be performed on the pieces because the can slide. otations can be performed on the pieces because each piece can be turned. eflections and glide reflections cannot be performed on the pieces because the back of a puzzle piece is useless in solving the puzzle. ilations are impossible because the pieces cannot change size... ample: glide reflectional, rotational, line, and translational. (-, ), M (, ), (, -). (, - ), M (,), (, - ) 0. (-, -), M (-, ), (0, 0). (, -), M (0, - ), (, -). Y Z = cm; X Z = cm; scale factor =. X Y = in.; Y Z = 0 in.; scale factor =. (-, ). (, ). (-, -). (, -). (, -). (-, ) hapter est, Form. Yes; the figures are the same shape and size.. o; the figures are not the same size.. Yes; the figures are the same shape and size.. (, ) ( -, + ). (, ) ( +, - ). (0, -), (-, 0), (-, -), (-, -). (, ), (0, 0), (, ), (, ). (, ), (-, ), (-, ), (0, 0). (, ), (, -), (-, ), (, ) 0. (-, 0), (0, -), (-, -), (-, -). (0, ), (, ), (, ), (, ). (, ) ( +, - ). (, ) ( -, + ). rhombus and two triangles; translation. ' ' ' ' '. 0 rotational smmetr. line smmetr. line smmetr; 0 rotational smmetr 0. (-, 0). (0, -). (-, ). (-, -). (, -). (-, ) lternative ssessment, Form K : coring Guide 0; 00 cm; all distances are magnified 0 times. tudent gives correct answers and a clear and accurate eplanation. tudent gives answers and an eplanation that are largel correct. tudent gives answers or an eplanation that ma contain serious errors. 0 tudent makes little or no effort. K : coring Guide ample: tudent draws a figure that accuratel reflects the stated conditions. tudent draws a figure that is mostl correct but falls just short of satisfing all the stated conditions. tudent draws a figure that has significant flaws relative to the stated conditions. 0 tudent makes little or no effort. Geometr hapter nswers

56 hapter nswers (continued) K : coring Guide ample: he car can undergo translations and rotations as it moves and turns. ut it can undergo neither a dilation, which would change its size, nor a reflection, which would, for eample, put the steering wheel on the passenger side. tudent gives a correct and thorough eplanation. tudent gives an eplanation that demonstrates understanding of transformations but ma contain minor errors or omissions. tudent gives an eplanation containing major errors or omissions. 0 tudent makes little or no effort. K : coring Guide ample: a. translation: b. dilation: c. reflection: translation vector:, 0 scale factor: line of reflection: = d. glide reflection: e. rotation: he glide reflection is equivalent to a translation, 0 followed b a reflection across the line =-. degrees of rotation: 0 tudent gives accurate graphs and correctl identifies the details the problem calls for. tudent gives generall correct graphs and details, although the work ma contain some errors. tudent gives graphs and accompaning details that contain significant errors or omissions. 0 tudent makes little or no effort. umulative eview.. G.. H.. J.. H. 0. J.. H.. J.. cm., H, I, M,,, U, V, W, X, Y. otational; it is turning around a fied point, the corner.. 0. quilateral triangle, square, and regular heagon; the measure of each interior angle is a factor of 0.. If a triangle is not a right triangle, then the thagorean heorem cannot be applied.. V V U U W W.. U G U G nswers Geometr hapter

57 hapter 0 nswers ractice ractice 0-. cm. in... ft. 0 m. 00 in.. ft. 0 cm.. in.. ft 0.. square units. " square units. " square units. cm.. in..,00 m. p. p. p ractice 0-..%. 0.%..% %..0%..%. 0%. 0%. 0%. % eteaching 0-. amples: ractice 0- " ". = ; =.". a = 0; c = ; d =. p = ";q = ; = 0. m = ; m = ; m = 0; m = 0. m = 0; m = 0; m = 0. m = 0; m = 0; m 0 = 0. "..". " 0. 0 in.. in.. in. ractice 0-. : ; :. : ; :. :;:. :. :. :. :. in. 00. cm 0. in.. :. :. :. 0 ft ractice 0-.. cm. ft. 0. mm.. m.. in... cm.. in.. cm.. m mi.. km.. in... mm..0 in... cm.. ft. 0. m.. m..0 ft 0.. ft. m.. ft ractice 0-. p. p..p. amples: F, F 0 0. amples: F, F. amples: F, 0 0. amples: F,.. F p in.. p cm. p m ractice 0-. p... p. p -.. p p p p or.. p or. -. p 0. p. p. p. p. p. amples:. 0 cm. in.. m. ft. in.. amples:. in. 0. cm. ft eteaching 0-. ample:. ample: bases = cm and 0 cm, height = cm. ample: cm. heck students work.. ample:. ample: diagonals = cm and cm. ample: cm. heck students work.. ft 0. in.. m.. cm Geometr hapter 0 nswers

58 hapter 0 nswers (continued) eteaching 0-.. ample: radius. ample: apothem =., radius =., side =. es. he are approimatel equal.. 00 in.. 00 ft. cm. m eteaching 0-. : ; :. : ; :. : ; :. :. :. :. :. :. : eteaching 0-.. in... cm.. in...0 in... in... in. a. in. b. area of square = s eteaching 0-. ample: p. ample:. ample: cm. ample: cm. cm. ample:. cm. he are approimatel p 0p 0p p equal.... p 0.. p p.. eteaching 0-.. ample: cm apothem. ample: 0 cm. ample: 0. cm. he are approimatel equal.. heck students work.. p ft. p in.. p m 0. 0.p ft. p cm. p in.. cm. cm eteaching 0-. heck students work.. %. heck students work.. ample: he are approimatel equal.. ample: he ratio would tend to be close to %.. % nrichment 0-. c; each segment is the hpotenuse of a right triangle with legs a and b.. o; the angles must be shown to be right angles.. he are congruent;. 0. 0; acute angles of a right triangle are complementar... m Q = 0 - (m + m ) = 0-0 = 0. epeat steps similar to those in ercises to show that Q, Q, and are right angles.. a + b; (a + b) = a + ab + b 0. c; c. a, b; ab. he areas of congruent triangles are equal.. a + ab + b = ( ab) + c or a + ab + b = ab + c. subtracting ab from both sides, ou obtain a + b = c (he thagorean heorem). nrichment 0-. hoose an interior point X; construct X, XQ, and X.. triangle is a trapezoid with one side where b = 0. Hence = h(b + b ) = hb.. rea of = rea - rea = [ h( + + b)] - [ h( + )] = h + h + hb - ( h + h) = hb nrichment 0-.. square units. square units. square units nrichment nrichment 0-. cm. " cm or about. cm. cm. = sin u, 0 u 0 nrichment 0-.. adius bject (r). Quarter. in.. ircle with a in. radius of in... iameter (d) ircumference (). he are approimatel equal.. p.. d. cm. cm. cm. in.. in.. in. p p heck student s work. heck student s work... 0 nswers Geometr hapter 0

59 hapter 0 nswers (continued) nrichment 0- " ". p. ".. p -. p.... p 0. ".. p - nrichment 0- ". 0.0 or or. 0. " hapter roject ctivit : Modeling in. ; 0. in. ; the seams in the finished quilt block account for the difference. lternative ssessment, Form K : coring Guide ample: a. b h b parallelogram with base b and height h has the same area as two triangles with base b and height h. hus, the area of the parallelogram is: = ( bh) = bh. b. b h ctivit : reating heck students work. ctivit : esearching heck students work. heckpoint Quiz.. m. 0 ft. m. in.. 00 ft. m. ; heckpoint Quiz.. ft.. m.. in... in.. 0. ft. 0p mi; p mi. p cm; p cm. p or about %. or 0% hapter 0 est, Form.. ft.. cm. ft. 0 square units. 0 square units. 0 square units cm.. in... ft.. in...0 cm. ample: raw an altitude to form a triangle. he altitude is ", and the area is "... cm. 0 square units. p. p 0. p. p. p.. square units..%..% hapter 0 est, Form. 0. ft.. cm ;. cm ; or. cm. 0. ft. 0 ft. cm. in in... cm.. cm. cm ; cm ; or cm. p. p. p. p.. cm. p cm 0. p in. b h trapezoid with bases b and b and height h has the same area as two triangles. he bases of the two triangles are b and b, and the heights of both are h. hus, the area of the trapezoid is: = b h + b h = h(b + b ). tudent gives accurate eplanations accompanied b clear and correct diagrams. tudent gives generall correct eplanations, although there ma be some errors in either the eplanations or the diagrams. tudent gives inaccurate or incomplete eplanations and diagrams. 0 tudent makes little or no attempt. K : coring Guide ample: (, ), (-, ). he slopes of and are equal, and the slopes of and are equal, so is a parallelogram. he slopes of and are opposite reciprocals, so #, and is a rectangle. = and =, so the area of is square units. tudent gives correct answers and provides complete and accurate reasoning. tudent gives generall correct answers and reasoning. tudent gives generall incorrect answers and reasoning. 0 tudent makes little or no attempt. K : coring Guide ample: a. QXY is a - -0 triangle, so the apothem QY = XY =, " X Y " or. " he perimeter is (XY) = ( ) = ". " he area is (XY) =? ( ) = ( )=. Q Geometr hapter 0 nswers

60 hapter 0 nswers (continued) b. forms a " triangle, so =, or. " he perimeter is () = " = ". he area is ap = ()( ") = ". tudent gives clear and correct diagrams and eplanations. tudent gives diagrams and eplanations that ma contain some minor errors. tudent gives diagrams or eplanations that contain major errors or omissions. 0 tudent makes little or no effort. K : coring Guide ample: a. a circle b. a and r are nearl equal; p and are nearl equal. c. If the number of sides is great enough that the polgon becomes indistinguishable from a circle, the two formulas are equivalent, and one can be derived from the other. For eample: = ap = r (r a and p ) = r(pr) ( = pr) = pr tudent gives correct answers to parts a and b and gives an adequate eplanation relating the two formulas in part c. tudent gives correct answers to parts a and b but ehibits difficult with the logic of part c. tudent gives answers and an eplanation that ma contain significant errors. 0 tudent makes little or no effort. umulative eview.. G.. J.. J.. H. 0. G.. J.. J.. J. ample: You ma have used the branches to climb into the tree.. ample: uppose that an equilateral triangle has an obtuse angle. hen one angle in the triangle has measure greater than 0. his contradicts the fact that, in an equilateral triangle, the measure of each angle is 0.. ample: cm cm cm cm 0. p - ".. m. uclid nswers Geometr hapter 0

61 hapter nswers ractice triangle. rectangle 0. rectangle. rectangle. rectangle ractice -.. cm.. ft.. cm.. m..0 ft.. cm a. 0 m b. 0 m a. in. b. in. a. mm b. mm 0a. cm 0b. 0 cm a. 00 m b. 00 m a. 000 ft b. 0 ft. p m..p cm. 0p ft ractice -. m. 0 cm. ft. p cm. 00p m. 00p in.. 0. m. 0.0 cm.. ft 0.. m.. cm.. ft ractice -..0 m.,. cm.. in... in... cm. 0. cm. 0 in.. ft. m 0. ft. cm. in.. ft. in.. m ractice -., cm. 00 in.. 0,0 in.. 00 d. 0 m. cm.. cm.. in... m 0.. in m.. ft. 0.. ractice -..0 in..,,. m.. cm.. m. 0. ft.. m.. mi.,0. cm.. m 0.. cm.,,. cm. 0,0. d. ft. m. cm. cm. cm. cm. cm ractice -. :. :. es; :. es; :. not similar. es; :. cm. in.. ft 0. ft. 0 m. 00 cm.,000 lb. lb eteaching -. ample: in. in. in. in. in. in. in. in. in. in. in. in. in. in. in.. heck students work.. vertices = ; faces = ; edges =. = V + F - ; =? + - ; =.... eteaching -.. heck students work.... = 00 cm ;.. = 0 cm.. cm. in... m eteaching -. ample: in. in. in. in. in. in. in. in.. heck students work.. ample: base = cm, slant height = cm. ample: cm. cm.. m.. in... ft eteaching -. 0 cm.. cm. m.. in.. ft.. cm eteaching cm. ft. in... m Geometr hapter nswers

62 hapter nswers (continued) eteaching -.. heck students work.. four circles. he are the same.. V =. in. ;.. =. in.. V =. cm ;.. =. cm. V =. m ;.. = 0. m 0. V =. ft ;.. =. ft. V = 0. in. ;.. =. in.. V = 0. mm ;.. =. mm eteaching -. :. :. :. :. 0 " :. :. : nrichment H U W FMU MHMII KW F HI IUI GMY, UU, UMI YI. nrichment -. in.. in.. in... in.. in.. in.. (s ) in.. ( - s ) in. 0.. ( 0.s ) in.. s in.. ( - 0.s) in.. in. s. in. ; no matter what the size of the base is, the surface area of the rectangular prism will alwas be in. as long as the entire piece of paper is used. nrichment -. a. b. trapezoid. l(a + b). l(a + b). a + b + l(a + b). π. pr. pl - prl 0. p( + r + l + rl). cm.. in.. 0. ft. m nrichment -. cubic units. 0 cubic units. cubic units. cubic units. 0 cubic units z (,, ) (0, 0, ) (0,, ) (, 0, ) (, 0, 0) (0,, 0) (,, 0) (0, 0, 0). cubic units (0,, ) (0,, 0) (, 0, 0) (, 0, ) nrichment -. on a line perpendicular to the base at the center of the square.,.. isosceles triangle. the height of the pramid. thagorean heorem. h =. s s. "l s "l s.,. V M s 0. slant height. apothem. height nas. h = "l a. = nas. "l a nrichment - a h h. pr. p. p(r - ).,000,000,000 mi. p(r - )(r + r + ). pd( ) = pd. act Volume pproimate d of hell Volume of hell ,0.,. 00,.,.. 0. in... in. z (0, 0, ) (0, 0, 0) nrichment -. 0 ft; 0 ft. $... in... in.. $.0.. in... in... in... in. 0.. in. nswers Geometr hapter

63 hapter nswers (continued) hapter roject ctivit : Measuring heck students work. ctivit : nalzing ample: l w d V. V :... b b ; b b ; the lower the ratio, the more material is being used to enclose the same volume.. to minimize the cost of packaging. ample: cereal bo is used for promoting the product, so it needs a large surface area on the front. ctivit : Investigating heck students work. ctivit : reating heck students work. heckpoint Quiz.. in. m. in.. 0p m..p in.... rectangle : : 0 : : in. m m in. in. in. in. in. heckpoint Quiz. 0. m ;. m.. ft ; 0. ft.. in. ; in... d ;. d. m ; m.. cm ; 0. cm.. in. hapter est, Form. ample:. ample: ft. he lateral area of a paint roller determines the amount of paint it can spread on a wall in one revolution. ecause the lateral area of roller is p in. and the lateral area of roller is p in., the both spread the same amount of paint in one revolution.. V = 00.0 cm ;.. = 0.0 cm. V =. in. ;.. =. in.. V = 00.0 cm ;.. =.0 cm. V = 00. ft ;.. =. ft. V =. m ;.. =. m. V = cm ;.. = cm 0a. a clinder with radius units and height units 0b. p cubic units 0c. p square units a. half of a sphere and a cone b.. cm. wo regular cans of soup each have a volume of. cm for a total of. cm. he single famil-size can with its volume of. cm is larger than the total volume of the two regular cans.. :. :..p cm... = 0. m ; V = 0. m... =. cm ; V =,0. cm. cm hapter est, Form. ample:. ample: cm ft ft m 0 m 0 cm in. in. cm cm. he lateral area of a paint roller determines the amount of paint it can spread on a wall in one revolution. ecause the lateral area of roller is p in. and the lateral area of roller is p in., roller spreads more paint on a wall in one revolution.. V =. m ;.. =. m. V =. cm ;.. =.0 cm. V =.0 ft ;.. =. ft. V = 0.0 in. ;.. =.0 in. a. half of a sphere and a cone b. 0. cm. wo regular cans of soup each have a volume of. cm for a total of. cm. he single famil-size can with its volume of. cm is larger than the total volume of the two regular cans =. m ; V = 0. m... =. cm ; V =,. cm Geometr hapter nswers

64 hapter nswers (continued) lternative ssessment, Form K : coring Guide ample: cm K : coring Guide ample: a. cm cm cm cm b. V 0 m ;.. 0 m rism:.. = 0 cm ; V = cm linder:.. = 0p cm or about. cm ; V = p cm or about. cm tudent gives accurate drawings and calculations. tudent gives drawings and calculations that ma contain minor errors. tudent gives drawings and calculations that contain significant errors or omissions. 0 tudent makes little or no attempt. K : coring Guide ample: ramids have polgonal sides, whereas cones have rounded sides. ach of these figures has a verte, and each has volume h. he base of a pramid is a polgon, and the base of a cone is a circle. h s a pramid with square base, a cone with radius r, height h, slant height s height h, and slant height s.. = + ( s).. = pr + prs = + s V = pr h V = h tudent gives accurate drawings and formulas. tudent gives drawings and formulas that ma contain minor errors. tudent gives drawings and formulas that contain significant errors. 0 tudent makes little or no attempt. h r s tudent draws an accurate net and gives correct answers. tudent draws a net and gives answers that ma contain minor errors. tudent draws a net and does calculations incorrectl. 0 tudent makes little or no attempt. K : coring Guide Foundation drawing: rthographic drawing: Front op ight tudent draws completel accurate figures. tudent draws generall accurate figures, but these ma contain a minor error. tudent draws figures containing significant errors. 0 tudent makes little or no attempt. umulative eview.. J.. J.. G.. H. 0. G.. F.. J. between d and d. clinders. p in.. ample: sharpened pencil is a composite of a cone and a clinder. he cone is the top of the pencil with the point, and the trunk of the pencil is the clinder.. heck students work. 0. oth shapes are triangles, and the sides are proportional in a : ratio. nswers Geometr hapter

65 hapter nswers ractice ".. ". "0. " 0. circumscribed. inscribed. circumscribed. cm. in.. ft ractice r = ; m.. r = ";m. # 0. r = ; m Q ; 0 0 U. J; K. onstruct radii and. F = because a diameter perpendicular to a chord bisects it, and chords and are congruent (given). hen, b H, F, and b, = F.. ecause congruent arcs have congruent chords, = =. hen, because an equilateral triangle is equiangular, m = m = m. ractice -. and ; and. and. and.. = ; = 0; z =. = 0; = = 0; = 0; z = 0 0. = 0; = 00; z = 0. = ; = ; z = 0. = 0; = 0; z = 0 a. m = 0 b. m = c. m = 0 d. m = a. m = 0 b. m = 0 c. m = 0 d. m = 00 ractice = ; = ; z =. = 0; =. = 0; = 0; z = 0 0. = ; =. = ; = ; z =. = 0; = ractice -. (0, 0), r =. (, ), r =. (-, -), r =. (-, ), r = ". + =. ( - ) + ( - ) =. ( - ) + ( - ) =. ( + ) + ( - ) =. ( + ) + ( + ) = 0. ( + ) + ( - ) =. + =. ( + ) + ( - ) =. + ( - ) =. ( - ) + ( + ) =. + ( + 0) = 00. ( + ) + ( + ) = (, ) (, ) (, ) (0, ) (0, 0) (, ). + =. ( - ) + ( - ) =. ( + ) + ( + ) =. ( - ) + ( + ) = 0. ( + ) + ( + ) =. ( - ) + ( - ) = (, 0) (, ) Geometr hapter nswers

66 hapter nswers (continued) ractice -.. cm Q.. cm in.. 0. in. 0. in F X H 0. in. 0. in. G J Y Q 0. in.. a third line parallel to the two given lines and midwa between them. a sphere whose center is the given point and whose radius is the given distance 0. the points within, but not on, a circle of radius in. centered at the given point 0. in... eteaching -. ". ". ".. eteaching - U mm mm 0. cm. ". ". ".. cm.. cm.. in... in. eteaching =, = eteaching = ; = 0; z =. = ; = ; z =. = ; = 0; z = eteaching -. ( - ) + ( - ) =. ( + ) + =. ( - ) + ( + ) =. + = 0. ( + ) + ( + ) =. ( - ) + ( - ) = 0. ( - ) + ( - ) =. ( - ) + ( - ) =. ( + ) + ( - ) = 0. (-, -); r =. (0, 0); r = 0.. (, 0); r = ". (, ); r = eteaching -.. heck students work.. the line perpendicular to at its midpoint. one, the midpoint of nswers Geometr hapter

67 hapter nswers (continued) nrichment -. Given. wo points determine a line segment.. wo tangents drawn to a circle from an eternal point are congruent.. adii of a circle are congruent.. radius and a tangent drawn to the same point of contact form a right angle.. efinition of a square. ides of a square are congruent.. ddition ropert. egment ddition ostulate 0. ubstitution ropert. ddition ropert. ubstitution ropert. ubtraction ropert nrichment - a. b. c. d. 0 e..,,, 0,. triangular numbers.. From each point, ( - ) chords ma be drawn. o, a total of ( - ) chords ma be drawn for points. ecause each chord has been drawn twice, divide b.. ( - ) nrichment -. Given. wo points determine a line segment.. he measures of eterior angles of a triangle equal the sum of the measures of the remote interior angles.. adii of a circle are congruent.. efinition of isosceles triangle. ase angles of an isosceles triangle are congruent.. efinition of congruent angles. ase angles of an isosceles triangle are congruent.. efinition of congruent angles 0. he measures of eterior angles of a triangle equal the sum of the measures of the remote interior angles.. ubstitution ropert. ivision ropert. drawing diameter passing through so that radius nrichment = 0; - = 0; =. =..... chords ), ;. secants, ;. tangent F and chord ; 0. chords ), ;. chords, ;. tangent F and secant F;. chords, ;. ) F, ; 00., ;. tangent F and chord ; nrichment - a. the edge of the fountain pool b. plaza ( ) c. fountain pool. a circle with center (-, ) and a radius of. all the interior points of a circle with center (, 0) and a radius of ". all points outside a circle with center (0, 0) and a radius of ". a circle and all its interior points with center (0, -) and a radius of. + = he sergeant can draw a circle with center (0, 0) and a radius of. he team will search in the circle. nrichment -.. heck students drawings.. with a radius of in.. two lines parallel to line l, one in. above l and the other in. below l. points. two lines parallel to line l, one in. above l and the other in. below l. points. two lines parallel to line l, one in. above l and the other in. below l. empt set. points. empt set hapter roject ctivit : oing heck students work. ctivit : ploring oth figures are made from circles with the same center. In Figure, the smallest circle and alternate rings are black. In Figure, vertical and horizontal tangents to the smallest circle are drawn. hen the tangents at angles to the first ones are drawn. lternating sections of the outer rings are black, as is the smallest ring. heck students work. ctivit : onstructing heck students work. heckpoint Quiz. in.. cm. 0 m..... =, = 00. =, =, z =. = 0 heckpoint Quiz. = 0. =, = 0. = 0, = 0. + ( - ) =.. ( + ) + ( - ) = hapter est, Form. (0, 0); r = 0. (, -); r =. (-, -); r = ". ( + ) + ( - ) =. ( - ) + ( - ) =. + ( - ) =. =.; =.. ( + ) + ( - ) = (, ) r Geometr hapter nswers

68 hapter nswers (continued) = ; = 0. = ; = ; w = ; z =. = 0; = ; z =. hapter est, Form 0 0. inscribed. circumscribed. ; Q. M;. center (0, 0); r =. center (, ); r = 0. + =. ( - ) + ( + ) =.. units;. square units 0. + ( + ) =. center (-, ); r = lternative ssessment, Form K : coring Guide (a) center (, ), radius = (b) he distance from point (, ) on a circle to the center (, ) is given b the formula = "( ) ( ). In an circle, the distance, or radius, is the same for all points on the circle. herefore the distance formula, when applied to the circle in this problem, becomes = "( ) ( ) for all points (, ) on the circle. quaring both sides of this equation ields the equation of the circle, = ( - ) + ( - ). tudent gives accurate answers and a correct eplanation. tudent gives answers or an eplanation that ma contain minor errors. tudent gives wrong answers or an incomplete or inaccurate eplanation. 0 tudent makes little or no effort. K : coring Guide (a) onstruct the perpendicular bisector of two chords. he point at which the meet is the center of the circle. (b) ecause the pentagon is regular, all chords are congruent. herefore the corresponding arcs are congruent. ecause there are 0 congruent arcs in the circle, each arc, and in particular, has measure. o find 0, call the center, and consider triangle. ecause has measure, so does angle. ecause =, is isosceles. herefore, the bisector of angle is perpendicular to (and bisects) chord, forming a - -0 triangle. hen sin = =, ielding =.0. nswers Geometr hapter

69 hapter nswers (continued) tudent devises a correct method and gives a correct answer and a valid eplanation. tudent devises a method, gives an answer, and gives an eplanation that ma contain some errors. tudent gives a method, an answer, and an eplanation that ma contain major errors or omissions. 0 tudent makes little or no effort. K : coring Guide (a) ngles and are inscribed angles. ach is inscribed in a different arc, but together the arc in which the are inscribed comprise the entire circle. herefore, m + m = (0) = 0. nd, because is a parallelogram, angles and are congruent. Finall, if angles are congruent and supplementar, then the are right. (b) he diagonals of bisect each other because is a parallelogram. uppose that the intersect at point X. hen X = X and X = X. lso, because is inscribed in a circle, the diagonals are chords, and therefore X? X = X? X. ubstituting ields X = X, and therefore X = X = X = X. hen =. (c) ither (a) or (b) allows ou to conclude that must be a rectangle. tudent gives valid and accurate arguments and eplanations. tudent gives arguments that, although basicall valid, ma contain minor flaws. tudent gives arguments containing major flaws. 0 tudent makes little or no attempt. K : coring Guide (a) ecause the circumscribing lines are tangent to the circle, = X, = Y, and X = Y. hen, because it is given that =, we have X = Y. dding segments together ields 0 =, showing that is isosceles. econdl, because XY has measure 00, XY has measure 0. hen m = (0-00) = 0, and because is isosceles, m = m = 0. Finall, = 0 because = = 0, and, b trigonometr, cos 0 =, 0 = cos 0 = 0. =.. nd again, =. (b) ecause angle has measure 0 and line is tangent at, triangle is a - -0 triangle. herefore, b trigonometr, tan =,= 0? tan =.. tudent gives accurate answers and eplanations. tudent gives answers and eplanations that ma contain minor errors. tudent gives answers and eplanations that contain significant errors. 0 tudent makes little or no attempt. umulative eview.. J.. F.. G.. F. 0. G.. H.. H.. ". heck students work.. tangent intersects a circle at eactl one point. secant intersects a circle at two points.. heck students work. 0. his follows from the thagorean heorem (c = a + b ) or the fact that the longest side of a triangle is opposite the largest angle.. If a triangle is not a right triangle, then it is acute.. Geometr hapter nswers

70 Geometr: ll-in-ne nswers Version hapter ractice -.,..0000,.0000.,. -,.,..,.. ample: or. or. or 0. or. or G. Y or. an heagon. circle with equall spaced diameters n(n ). a. angle. hand-shakes. h =.. ample: he farther out ou go, the closer the ratio gets to a number that is approimatel ,,,,,,,.. ight Front Front op ight Guided roblem olving -. he pattern is easier to visualize.. he graph will go up.. Use Years for the horizontal ais.. Use umber of tations for the vertical ais.. increasing. greater. Yes. ince the number of stations increases steadil from 0 to 000, we can be confident that the number of stations in 00 will be greater than in atterns are necessar to reach a conclusion through inductive reasoning.. (an list of numbers without a pattern would appl),;,;,;,; 0, ractice Front Front Front ight Front ight ight ight Front ight.,, Guided roblem olving -. he represent three-dimensional objects on a twodimensional surface.. nine. ee the figure in, below... Yes. It is similar to the foundation drawing, ecept there are no numbers.. no. op. Yes.. Front op Front ight ight Front op ight ractice -. * ). an two of the following:,,,,,,,. oints,, and are collinear.. es. es. no. no. es. no ll-in-ne nswers Version Geometr

71 Geometr: ll-in-ne nswers Version (continued) 0. es. * es ). es * ). no. es. es. G... the empt set 0. *K ) M. M. ample: plane. ample: plane. * ). es. no. es. the empt set. no 0. es. es Guided roblem olving -. ollinear points lie on the same line.. nswers ma var. (ome people might note that the -coordinate of two of the points is the same so that the third point must have the same -coordinate to be collinear. ince it does not, the points are not collinear.). horizontal. o.. o.. ll points must have the same -coordinate, -.. o.. a, b ractice -. true. false. true. false. false. false.,.. an three of the following pairs: and ; and ; and ; and ; and * * ) ) * * ) ) * ) ; * and ) * * ) * ) * ) ) * ) ; * and ) * * ) * ; ) * ) ) * ) and ; and ; *G ) * ) * ) *J ) * ) *FK ) * ) JK HG H F JH F GK HG J HG JK H JK FG J FG H FK KG H KG JH KF JH and ; and ; and 0. planes and. planes and ; planes and. planes and. ) planes ) and ). ample: ) ) ). ). ) ) G F and or G and. F, F. GF, G. es 0. ample: U. ample: Guided roblem olving - Q. pposite ras are two collinear ras with the same endpoint.. a line. ee graph in ercise answer. W. nswers ma var. ample: (0, 0) (nswers will be coordinates (, ), where =, <.) (0,0). es. (, ) ractice or. - or. ; ; no. ; ; es 0. or = ; = ; =. = ; =0; =0 Guided roblem olving -.. =. egment ddition ostulate.. ince =, = ().. = () =. =. = =. nswers ma var.. =, =, = ractice -. an three of the following: &, &M, &M, &. &. &. & &, &, &F, &F, &F. &F, &. &, &. = ; ; ;. = ; ; ; 0 Guided roblem olving -. ngle ddition ostulate. supplementar angles. m&q + m&q = 0. ( + ) + ( + 0) = 0. =.. m&q = ; m&q =. he sum of the angle measures should be 0; m&q + m&q = + = 0. a. = b. m& = ; m& = ractice -. (,) (,) X Y Geometr ll-in-ne nswers Version

72 Geometr: ll-in-ne nswers Version (continued).. X Y 0.? X K M Y X Z 0 M..? X Z. true. false. false. true. true. false 0. true Guided roblem olving -. & > &. complementar angles. &. m& = m& =. m& = m& + m& = + =. m& + m& = 0 m& + = 0 m& =. m&f = m& =. nswers ma var. ample: he sum of the measures of the complementar angles should be 0 and the sum of the measures of the supplementar angles should be 0.. m&=, m&f =, m& =, and m& = W X X ll-in-ne nswers Version Geometr

73 Geometr: ll-in-ne nswers Version (continued) ractice -.. Guided roblem olving -. si. It is a two-dimensional pattern ou can fold to form a three-dimensional object.. rectangles.. ".. " ".. (, ). (, ). (0, -). (-, ). (-0.,.). (, - ). (, -). (, 0) 0. (-, ). es; = = = =. " Q... (., ) Guided roblem olving -. istance Formula. he distance d between two points (, ) to (, ) is d Í ( ) ( ).. o; the differences are opposites but the squares of the differences are the same.. XY Í ( ()) ( ). o the nearest tenth, XY =. units.. o the nearest tenth, XZ =.0 units.. Z is closer to X.. he results are the same; e.g., XY Í ( ) ( ()) Í, or about. units, as before.. YZ Í ( ()) ( ) Í; to the nearest tenth, YZ =. units. o the nearest tenth, XY+YZ+XZ =. units. ractice -. in.. 0 in... m. p. p..p. cm; cm. cm; cm. in.; in ft; ft. 0 m; m. 0 m; m. ;. ;. 0;..p p. 0,000p.. ; 0.. ;. 0 in.. he are equal.. 0 in.. nswers will var. ample: ( ) + ( ) + ( ); the results are the same, 0 in.. ( ) = in. : Graphic rganizer. ools of Geometr. nswers ma var. ample: patterns and inductive reasoning; measuring segments and angles; basic constructions; and the coordinate plane. heck students work. : eading omprehension * ) * ) * ) * ). nswer ma var. ample:, F GH, JK > M, J > KM, /HK > /KM, * ) * ) ) * ' * ) ), HI, m/jf + m/fjk = 0,. oints, M, and are collinear.. and intersect at point M.. a : eading/writing Math mbols. ine is parallel to line M.. ine. ine segment GH. a. he length of segment XY is greater than the length of segment.. M = XY. GH = (K). ' UV. plane plane XYZ 0. : Visual Vocabular ractice. parallel planes. egment ddition ostulate. supplementar angles. opposite ras. isometric drawing. perpendicular lines. foundation drawing. right angle. congruent sides : Vocabular heck et: two-dimensional pattern that ou can fold to form a three-dimensional figure. onjecture: conclusion reached using inductive reasoning. ollinear points: oints that lie on the same line. Midpoint: point that divides a line segment into two congruent segments. ostulate: n accepted statement of fact. Geometr ll-in-ne nswers Version

74 Geometr: ll-in-ne nswers Version (continued) F: Vocabular eview uzzle I Z J Y J Z H G K F V I I U X W Z G W I Y I Y Y U G F Q G G K I I M I U X F V V U J V K Y H U I V H X I I X G M X G M M I V I X K F K V F J hapter ractice -. ample: It is :00 noon on a rain da.. ample: he car will not start because of a dead batter.. ample:. If ou are strong, then ou drink milk.. If a rectangle is a square, then it has four sides the same length.. If ou are tired, then ou did not sleep.. If =, then - = ; true.. If 0, then 0; true.. If m is positive, then m is positive; true. 0. If - =, then = ; true.. If 0, then point is in the first quadrant; false; (, -) is in the fourth quadrant.. If lines are parallel, then their slopes are equal; true.. If ou have a sibling, then ou are a twin; false; a brother or sister born on a different da.. siblings twin G. Hpothesis: If ou like to shop; conclusion: Visit igeon Forge outlets in ennessee.. igeon Forge outlets are good for shopping.. If ou visit igeon Forge outlets, then ou like to shop.. It is not necessaril true. eople ma go to igeon Forge outlets because the people the are with want to go there.. rinking ustain makes ou train harder and run faster. 0. If ou drink ustain, then ou will train harder and run faster.. If ou train harder and run faster, then ou drink ustain. Guided roblem olving -. Hpothesis: is an integer divisible b.. onclusion: is an integer divisible b.. Yes, it is true. ince is a factor U I G V I V G J I H K U I W V J G W K I K Q M W F U W U U I F U Q V Q K Z H I F of, it must be a factor of? =.. If is an integer divisible b then is an integer divisible b.. he converse is false. ountereamples ma var. et =. hen Í, which is not an integer and is not divisible b.. o. he conditional is true, so there is no such countereample.. o. definition, a general statement is false if a countereample can be provided.. If + =, then =. he original statement and the converse are both true. ractice -. wo angles have the same measure if and onl if the are congruent.. - = if and onl if =.. he converse, If n =, then n =, is not true.. figure has eight sides if and onl if it is an octagon.. If a whole number is a multiple of, then its last digit is either 0 or. If a whole number has a last digit of 0 or, then it is a multiple of.. If two lines are perpendicular, then the lines form four right angles. If two lines form four right angles, then the lines are perpendicular.. If ou live in eas, then ou live in the largest state in the contiguous United tates. If ou live in the largest state in the contiguous United tates, then ou live in eas.. ample: ther vehicles, such as trucks, fit this description.. ample: ther objects, such as spheres, are round. 0. ample: his is not specific enough; man numbers in a set could fit this description.. ample: aseball also fits this definition.. ample: leasing, smooth, and rigid all are too vague.. es. no. es Guided roblem olving -. good definition is clearl understood, precise, and reversible.. & and &, & and &. o ll-in-ne nswers Version Geometr

75 Geometr: ll-in-ne nswers Version (continued). he are not supplementar.. linear pair has a common verte, shares a common side, and is supplementar.. es. es; es. linear pairs: & and &, & and &; not linear pairs: & and &, & and &, & and &, & and &. (0,) (,) ractice -. & and & are complementar.. Football practice is canceled for Monda.. F is a right triangle.. If ou liked the movie, then ou enjoed ourself.. If two lines are not parallel, then the intersect at a point.. If ou vacation at the beach, then ou like Florida.. not possible. amika lives in ebraska.. not possible 0. It is not freezing outside.. hannon lives in the smallest state in the United tates.. n hursda, the track team warms up b jogging miles. Guided roblem olving -. conditional; hpothesis. Yes. eth will go.. nita, eth, isha, amon. o; onl two students went.. eth, isha, amon; no onl two went.. isha, amon. he answer is reasonable. It is not possible for another pair to go to the concert.. amon ractice -. U = M. m&qw = 0. = M. =. J. Given; ddition ropert of qualit; ivision ropert of qualit. Given; ddition; ubtraction ropert of qualit; Multiplication ropert of qualit; ivision ropert of qualit. ubstitution. ubstitution 0. ransitive ropert of qualit. mmetric ropert of ongruence. ransitive ropert of ongruence. efinition of omplementar ngles; 0, ubstitution;, implif;,, ubtraction ropert of qualit;, ivision ropert of qualit. Given; ( - ), ubstitution; -, istributive ropert; -, ubtraction ropert of qualit;, Multiplication ropert of qualit Guided roblem olving -. ngle ddition ostulate. ubstitution ropert of qualit; implif; ddition ropert of qualit; ivision ropert of qualit. 0. es; es. ; ractice m& = ; m& =. m& = ; m& =. m& =, m& = 0. m& = 0; m& = 0. m&m = ; m&mq = ; m&qm =. m& = m& = 0; m& = m& = ; m& = m& =. m&w = m&w; m&w + m&w = 0; m&w + m&w = 0; m&w = m&w Guided roblem olving ee graph in ercise answer.. on the positive -ais. nswers ma var. can be an point on the positive -ais. ample: (0, ). (0,0) (, ) X(,0). he are adjacent complementar angles.. nswers ma var. can be an point on the line,. 0. ample: (, -).. a right angle. Yes; their sum corresponds to the right angle formed b the positive -ais and the positive -ais. 0. nswers ma var. can be an point on the negative -ais, sample: (-, 0) : Graphic rganizer. easoning and roof. nswers ma var. ample: conditional statements; writing biconditionals; converses; and using the aw of etachment. heck students work. : eading omprehension. degrees. degrees. b : eading/writing Math mbols. egment M is congruent to segment Q.. If p, then q.. he length of M is equal to the length of Q.. ngle XQV is congruent to angle.. If q, then p.. he measure of angle XQV is equal to the measure of angle.. p if and onl if q.. a b. = M 0. m/xyz = m/. b a. > M. a b. /XYZ / : Visual Vocabular ractice. aw of etachment. hpothesis. istributive ropert. efleive ropert. aw of llogism. biconditional. conclusion. good definition. mmetric ropert : Vocabular heck ruth value: rue or false according to whether the statement is true or false, respectivel Hpothesis: he part that follows if in an if-then statement. iconditional: he combination of a conditional statement and its converse; it contains the words if and onl if. onclusion: he part of an if-then statement that follows then. onditional: n if-then statement. Geometr ll-in-ne nswers Version

76 Geometr: ll-in-ne nswers Version (continued) F: Vocabular eview uzzle 0 hapter U ractice -. corresponding angles. alternate interior angles. same-side interior angles. alternate interior angles. same-side interior angles. corresponding angles. and, and, and, and. and, and. and, and 0. m = 00, alternate interior angles; m = 00, corresponding angles or vertical angles. m =, alternate interior angles; m =, vertical angles or corresponding angles. m =, corresponding angles; m =, vertical angles. = 0;, 0. = ;,. = 0;, a. lternate Interior ngles heorem b. Vertical angles are congruent. c. ransitive ropert of ongruence Guided roblem olving - I G J U I U G M U I G H Y H I I. he top and bottom sides are parallel, and the left and right sides are parallel.. he two diagonals are transversals, and also each side of the parallelogram is a transversal for the two sides adjacent to it.. orresponding angles, interior and eterior angles are formed.. v, w and ; the lternate Interior ngles heorem, v =, w = and =.. nswers ma var. ossible answer: the ame-ide Interior ngles heorem, (w + ) + ( + ) = 0. ince w =, =. (he two s are equal b heorem -.). w =, =, v =, = ; es. v =, w =, =, = J V I I I ractice - * ) a. same-side interior b. c. * ) Q d. same-side interior e. ame-ide Interior ngles f. * ) g. -. l and m, onverse of ame-ide Interior ngles heorem. none. and, onverse of ame-ide Interior ngles heorem. and HU, onverse of orresponding ngles ostulate. H and I, onverse of orresponding ngles ostulate. a and b, onverse of ame-ide Interior ngles heorem Guided roblem olving -. and m. transversals.. the angles measuring º and º. 0º. º. 0 - = or + = 0. =. With =, = and =. 0. = ractice -. rue. ver avenue will be parallel to Founders venue, and therefore ever avenue will be perpendicular to enter it oulevard, and therefore ever avenue will be perpendicular to an street that is parallel to enter it oulevard.. ot necessaril true. o information has been given about the spacing of the streets.. rue. he fact that one intersection is perpendicular, plus the fact that ever street belongs to one of two groups of parallel streets, is enough to guarantee that all intersections are perpendicular.. rue. pposite sides of each block must be of the same tpe (avenue ll-in-ne nswers Version Geometr

77 Geometr: ll-in-ne nswers Version (continued) or boulevard), and adjacent sides must be of opposite tpe.. ot necessaril true. If there are more than three avenues and more than three boulevards, there will be some blocks bordered b neither enter it oulevard nor Founders venue.. a e. a e. a e. a e 0. a e. a e. If the number of statements is even, then n. If it is odd, then n.. he proof can instead use alternate interior angles or alternate eterior angles (if the angles are congruent, the lines are parallel) or same-side interior or same-side eterior angles (if the angles are supplementar, the lines are parallel).. It is possible. Guided roblem olving -. supplementar angles. right angle. n one of the following: ostulate -, or heorem -, -, - or It is congruent; ostulate a c. It is true for an line parallel to b.. Yes. he point is that a transversal cannot be perpendicular to just one of two parallel lines. It has to be perpendicular to both, or else to neither. ractice = ; = ; z =. a = ; b = ; c =. v = ; w = ; t = m = ; m = ; m =.... m = ; m =. isosceles. obtuse scalene. right scalene. obtuse isosceles 0. equiangular equilateral Guided roblem olving -. three. 0. right triangle. z = 0; ecause it is given in the figure that.. heorem -, the riangle ngle-um heorem. =. =. # is a --0 right triangle. # is a --0 right triangle.. 0. # is a -- acute triangle.. Yes, all three are acute angles, with & visibl larger than & and &.. & ractice -. = 0; = 0. n =. a = 0; b = F. F and.. 0 l l l Guided roblem olving -. theater stage, consisting of a large platform surrounding a smaller platform. he shapes in the bottom part of the figure ma represent a ramp for actors to enter and eit.. he measures of angles and. ; octagon. ( - )0 = 00 degrees... Yes, angle is an obtuse angle and angle is an acute angle.. trapezoids. 0 ractice -. = -. =- +. = -. =- -. = -. = -. = -. = Geometr ll-in-ne nswers Version

78 Geometr: ll-in-ne nswers Version (continued) = +. = +. = -. =- -. = + 0. = +. =- -. =- +. = ; =-. = 0; =. =-; =-. =-; =.. (, 0) 0 (0, ). 0. (0, ) (, 0).. (, 0) (0, ). (, 0) (, 0) (0, ) (0, ) (0, ) (, 0).. a. m = $0.0 b. the amount of mone the worker is paid for each bo loaded onto the truck c. b = $.0 d. the base amount the worker is paid per hour. = - + Guided roblem olving -. (, 0) 0. m. - = m(- ). lope of å å = ; slope of = -. he absolute values of the slopes are the same, but one slope is positive and the other is negative.. oint-slope form: - 0 = ( - 0); slope-intercept form: =. oint-slope form: - =- ( - ) or - 0 =- ( - ); å slope-intercept form: = f line : (0, 0); å of line : (0, 0). # appears to be an isosceles triangle, which is consistent with a horizontal base and two remiaining sides having slopes of equal magnitude and opposite sign.. lope = 0; = 0; -intercept = (0, 0) just as for line å (the intersect on the -ais). ractice - (0, ). neither;,? -. perpendicular;? - =-. parallel; - = -. parallel; - =-. perpendicular; = is a horizontal line, = 0 is a vertical line. parallel; - = -. neither;,? -. parallel; - = -. perpendicular; -? =- 0. neither; -,? -. neither; - -, -? - - (0, ) (, 0) ll-in-ne nswers Version Geometr

79 Geometr: ll-in-ne nswers Version (continued). neither; -,? - -. neither;,? -. parallel; =. =. =- +. =- -. = +. = 0 0. = -. = 0. ample:. ample: b a Guided roblem olving -.. a right angle. m m =-. sides GH and GK. lope of GH = ; slope of GK =-. roduct =- -. ides GH and GK are not perpendicular.. #GHK has no pair of perpendicular sides. It is not a right triangle.. o. &HGK; approimatel 0 0. lope of M = and slope of =-. he product of the slopes is -, so M and are perpendicular. ractice K Q K G H.. a. ample: b b c b c b b b c Geometr ll-in-ne nswers Version

80 Geometr: ll-in-ne nswers Version (continued).. ine is perpendicular to line F.. ngle and angle are complementar. 0. ngle is a right angle, or the measure of angle is 0.. ample answer: å å G, m/f = m/gf a a : Visual Vocabular ractice/high-use cademic Words. propert. conclusion. describe. formula. measure. approimate. compare. contradiction. pattern Guided roblem olving -. a line segment of length c. onstruct a quadrilateral with one pair of parallel sides of length c, and then eamine the other pair.. he procedure is given on p. of the tet.. djust the compass to eactl span line segment c, end to end. hen tighten down the compass adjustment as necessar.. m. m l. he appear to be both congruent and parallel.. he answers to tep are confirmed.. es; a parallelogram : Graphic rganizer. arallel and erpendicular ines. nswers ma var. ample: properties of parallel lines; finding the measures of angles in triangles; classifing polgons; and graphing lines. heck students work. : eading omprehension. spaces. spaces. 0. corresponding. $000; $0. the width of the stalls, 0 ft. b : eading/writing Math mbols l m l å å. m # n. m/ + m/ = 0.. m/m + m/mq = 0. / /F. ine is parallel to line.. he measure of angle is equal to the measure of angle XYZ. : Vocabular heck ransversal: line that intersects two coplanar lines in two points. lternate interior angles: onadjacent interior angles that lie on opposite sides of the transversal. ame-side interior angles: Interior angles that lie on the same side of a transversal between two lines. orresponding angles: ngles that lie on the same side of a transversal between two lines, in corresponding positions. Flow proof: convincing argument that uses deductive reasoning, in which arrows show the logical connections between the statements. F: Vocabular eview F. K. H. 0. G. I. J hapter ractice -. m = 0; m = 0. m = 0; m =. m = 0; m = 0; m = 0; m = 0. J,, J. J,,. WZ JM, WX JK, XY K, ZY M. W J, X K, Y, Z M. Yes; GHJ IHJ b heorem - and b the efleive ropert of. herefore, GHJ IHJ b the definition of triangles.. o; Q V because vertical angles are congruent, and Q V b heorem -, but none of the sides are necessaril congruent. 0a. Given 0b. Vertical angles are. 0c. heorem - 0d. Given 0e. efinition of triangles Guided roblem olving -. right triangles. m& =, m& = m& = 0, and = in.. # is congruent to #KM means corresponding sides and angles are congruent.. and t.. m&k = m&m =. =. =. 0. t =. t =. he angle measures indicate that the two triangles are isosceles right triangles. his matches the appearance of the figure.. m&m = 0 ll-in-ne nswers Version Geometr

81 Geometr: ll-in-ne nswers Version (continued) ractice -. b. not possible. not possible. U XWV b. not possible. GHF b. MK KMJ b. Q b. not possible 0.. and. and. a. Given b. efleive ropert of ongruence c. ostulate. tatements easons. F FG, F FH. Given. F HFG. Vertical s are.. F HFG. ostulate Guided roblem olving -. I > and bisects &I.. rove whatever additional facts can be proven about #I and #, based on the given information.. I >. &I &.. #I # b ostulate -, the ide-ngle- ide () ostulate. It does not matter. he ide-ngle- ide ostulate applies whether or not the are collinear.. It does follow, because #I # and because I and Q are corresponding parts. ractice -. not possible. ostulate. heorem. heorem. not possible. not possible. ostulate. not possible. heorem 0. tatements easons. K M, K M. Given. JK M. Vertical s are.. JK M. ostulate. Q Given Q Given efleive ropert of. F. KHJ HKG or KJH HGK. M Q Q heorem Guided roblem olving -. corresponding angles and alternate interior angles. & and &. & and &.. & and &. & > &, >, and & > &. # > # b ostulate -, the ngle- ide-ngle () ostulate. Yes; heorem -, the ngle-ngle-ide () heorem; & > &. o, because now there is no wa to demonstrate a second pair of congruent sides, nor a second pair of congruent angles. ractice -. is a common side, so b, and b.. FH is a common side, so FH HFG b, and H FG b.. KJ M b, so K b.. Q is a common side, so Q Q b. Q Q b.. VX is a common side, so UVX WVX b, and U W b.. ZY and are vertical angles, so YZ b. Z b.. G is a common side, so G FG b, and FG G b.. JKH and KM are vertical angles, so HJK MK b, and JK K b.. is a common side, so Q b, and Q b. 0. First, show that and are vertical angles. hen, show b. ast, show b.. First, show FH as a common side. hen, show JFH GHF b. ast, show FG JH b. Guided roblem olving -. compass with a fied setting was used to draw two circular arcs, both centered at point but crossing in different locations, which were labeled and. he compass was used again, with a wider setting, to draw two intersecting circular arcs, one centered at and one at. he point at å which the new arcs intersected was labeled. Finall, line was drawn.. Find equal lengths or distances and eplain å wh is perpendicular to.. # and #. = and =.. # > #,b ostulate -, the ide-ide-ide () ostulate. & > & b. ince & > &, m& = m& and m& + m& = 0, it follows that m& = m& = 0.. from the definition of perpendicular and the fact that m& = m& = 0. he distances do not matter, so long as = and =. hat is what is required in order that # > #. 0. raw a line, and use the construction technique of the problem to construct a second line perpendicular to the first. hen do the same thing again to construct a third line perpendicular to the second line. he first and third lines will be parallel, b heorem -0. ractice -. = ; =. = 0; = 0. t = 0. r = ; s =. = ; = 0; z =. a = ; b = ; c = 0. =. a = 0; b = 0; c =. z = 0 0. ; F. G; G G. KJ; KIJ KJI. ;. ; J J. ; H H = 0; =. = 0; = 0. = ; = Geometr ll-in-ne nswers Version

82 Geometr: ll-in-ne nswers Version (continued) Guided roblem olving -. ne angle is obtuse. he other two angles are acute and congruent.. Highlight an obtuse isosceles angle and find its angle measures, then find all the other angle measures represented in the figure.. ossible answer:. GI JI Given HI HI efleive ropert of IHG IHJ H heorem. 0. 0, because the measure of each base angle is half the measure of an angle of the equilateral triangle.. 0 because the sum of the angles of the highlighted triangle must equal 0.. he other measures are 0 and 0. amples:. Yes; 0 = 0.. nswers ma var. ractice -. tatements easons. #, # F. Given., are right s.. erpendicular lines form right s.. F,. Given. F. H heorem. tatements easons., are right s.. Given. Q. Given. Q Q. efleive ropert of. Q Q. H heorem. MJ and MJK are right s. erpendicular lines form right s. GHI and JHI GHI JHI are right s. Given heorem - m GHI + m JHI = 0 ngle ddition ostulate. VW. none. m and m F = 0. GH JH. 0. UV or V U. m and m X = 0. m F and m = 0. GI # JH Guided roblem olving -. wo congruent right triangles. ach one has a leg and a hpotenuse labeled with a variable epression.. the values of and for which the triangles are congruent b H. he two shorter legs are congruent.. = +. he hpotenuses are congruent.. + =. = ; = hpotenuse. shorter leg = ; es, this matches the figure.. he solution remains the same: = and =. he reason is that one is still solving the same two equations, = + and + =. ractice -. ZWX YXW;. ;. JG FKH;. M;. F G;. UVY VUX;. common side:. F G G F common side: FG. common angle: K H M MK Given MJ MJK H heorem M J MJ MJ efleive ropert of ll-in-ne nswers Version Geometr

83 Geometr: ll-in-ne nswers Version (continued) 0. ample: tatements easons. X Y. Given. X #, Y #. Given. m X and. erpendicular lines m Y = 0 form right s... efleive ropert of. Y X. ostulate. ample: ecause FH G, HFG GF, and FG GF, then FG GFH b. hus, F GH b and H H, then GH FH b. Guided roblem olving -. he figure, a list of parallel and perpendicular pairs of sides, and one known angle measure, namel m& =. nine. he are congruent and have equal measures.. m& = m& = m& =. m& = 0. m& =. m& =. m& =. m&fg = 0 0. m& =. m& =, m& =, and m& =. m& = + =. m&fi = 0 -(m& + m&) = 0; m&h = m& = 0; m&fj = 0 - m& = ; m&ig = m&fi = 0 : Graphic rganizer. ongruent riangles. nswers ma var. ample: congruent figures; triangle congruence b,,, and ; proving parts of triangles congruent; the Isosceles riangle heorem. heck students work. : eading omprehension. Yes. Using the Isosceles riangle heorem, /W /Y. It is given that WX YX and WU YV. herefore nwux nyvx b.. here is not enough information. You need to know if, if / /,or if / /.. a : eading/writing Math mbols. ngle-ngle-ide. triangle XYZ. angle Q. line segment. line. ra WX. hpotenuseleg. line. angle 0. ngle-ide-ngle : Visual Vocabular ractice. theorem. congruent polgons. base angle of an isosceles triangle.. postulate. verte angle of an isosceles triangle. corollar. base of an isosceles triangle. legs of an isosceles triangle : Vocabular heck ngle: Formed b two ras with the same endpoint. ongruent angles: ngles that have the same measure. ongruent segments: egments that have the same length. orresponding polgons: olgons that have corresponding sides congruent and corresponding angles congruent. : n abbreviation for corresponding parts of congruent triangles are congruent. F: Vocabular eview uzzle. postulate. hpotenuse. angle. verte. side. leg. perpendicular. polgon. supplementar 0. parallel. corresponding hapter ractice - a. cm b. cm c. cm a.. in. b.. in. c.. in. a.. cm b.. cm c.. cm a. 0b.. GH, HI, GI. YZ, Q XZ, XY Q Guided roblem olving -. 0 units. he three sides of the large triangle are each bisected b intersections with the two line segments ling in the interior of the large triangle.. the value of. he are called midsegments.. he are parallel, and the side labeled 0 is half the length of the side labeled.. = 0. Yes; the side labeled appears to be about twice as long as the side labeled 0.. o. hose lengths are not fied b the given information. (he triangle could be verticall stretched or shrunk without changing the lengths of the labeled sides.) ll one can sa is that the midsegment is half as long as the side it is parallel to. ractice -. WY is the perpendicular bisector of XZ right triangle.... equidistant 0. isosceles triangle )..... right triangle. J is the bisector of J.. ) ample: oint M lies on J.. right isosceles triangle Geometr ll-in-ne nswers Version

84 Geometr: ll-in-ne nswers Version (continued) Guided roblem olving -.. ee answer to tep, above.. ee answer to tep, above.. lot a point and eplain wh it lies on the bisector of the angle at the origin.. line : ; line m: = 0. (0, ). = = ; es. heorem -, the onverse of the ngle isector heorem. m& = m& < 0. raw, m, and, then draw. ince = = 0, it follows that # > #, b H. hen > and & > & b. ractice -. (-, ). (, 0). (, ). heck students work. he final result of the construction is shown. <. altitude. median. none of these. perpendicular bisector. angle bisector 0. altitude a. (, 0) b. (-, -) a. (0, 0) b. (, -) a. (0, 0) b. (0, ) Guided roblem olving - m. the figure and a proof with some parts left blank. Fill in the blanks... heorem -, the erpendicular isector heorem. ; X. the ransitive ropert of qualit. erpendicular isector. (his converse is heorem -.). he point of the proof is to demonstrate that n runs through point X. It would not be appropriate to show that fact as alread given in the figure.. othing essential would change. oint X would lie outside # (below å ), but the proof woud run just the same. ractice -. I and III. I and II. he angle measure is not.. ina does not have her driver s license.. he figure does not have eight sides.. he restaurant is open on unda.. is congruent to XYZ.. m Y 0 a. If two triangles are not congruent, then their corresponding angles are not congruent; false. b. If corresponding angles are not congruent, then the triangles are not congruent; true. 0a. If ou do not live in oronto, then ou do not live in anada; false. 0b. If ou do not live in anada, then ou do not live in oronto; true.. ssume that m m.. ssume that UVW is not a trapezoid.. ssume that M does not intersect.. ssume that FGH is not equilateral.. ssume that it is not sunn outside.. ssume that is obtuse.. ssume that m 0. his means that m + m 0. his, in turn, means that the sum of the angles of eceeds 0, which contradicts the riangle ngle-um heorem. o the assumption that m 0 must be incorrect. herefore, m 0. Guided roblem olving -. Ice is forming on the sidewalk in front of oni s house.. Use indirect reasoning to show that the temperature of the sidewalk surface must be F or lower.. he temperature of the sidewalk in front of oni s house is greater than F.. Water is liquid (ice does not form) above F.. here is no ice forming on the sidewalk in front of oni s house.. he result from tep contradicts the information identified as given in tep.. he temperature of the sidewalk in front of oni s house is less than or equal to F.. If the temperature is above F, water remains liquid. his is reliabl true. onverse: If water remains liquid, the temperature is above F. his is not reliabl true. dding salt will cause water to remain liquid even below F.. uppose two people are each the world s tallest person. all them person and person. hen person would be taller than everone else, including, but b the same token would be taller than. It is a contradiction for two people each to be taller than the other. o it is impossible for two people each to be the World s allest erson. ractice -. M,.,., Q.,.,.,. es; +, +, +. no; + 0. es; + 0. es; +, +, +. es; +, +, +. no; no; no; +. no;. +..,,.,,.,,.,, 0.,, J.,, Guided roblem olving -. ll-in-ne nswers Version Geometr

85 Geometr: ll-in-ne nswers Version (continued). he side opposite the larger included angle is greater than the side opposite the smaller included angle.. he angle opposite the larger side is greater than the angle opposite the smaller side. he opposite sides each have a length of nearl the sum of the other two side lengths.. he opposite sides are the same length. he are corresponding parts of triangles that are congruent b. : Graphic rganizer. elationships Within riangles. nswers ma var. ample: midsegments of triangles; bisectors in triangles; concurrent lines, medians, and altitudes; and inverses, contrapositives, and indirect reasoning. heck students work. : eading omprehension. he width of the tar pit is 0 meters.. b : eading/writing Math mbols.. F.. G.. I. M. H. 0. K.... J. : Visual Vocabular ractice. median. negation. circumcenter. contrapositive. centroid. equivalent statements. incenter. inverse. altitude : Vocabular heck Midpoint: point that divides a line segment into two congruent segments. Midsegment of a triangle: he segment that joins the midpoints of two sides of a triangle. roof: convincing argument that uses deductive reasoning. oordinate proof: proof in which a figure is drawn on a coordinate plane and the formulas for slope, midpoint, and distance are used to prove properties of the figure. istance from a point to a line: he length of the perpendicular segment from the point to the line. F: Vocabular eview. altitude. line. median. negation. contrapositive. incenter. orthocenter. slope-intercept. eterior 0. obtuse. alternate interior. centroid. equivalent. right. parallel hapter ractice -. parallelogram. rectangle. quadrilateral. parallelogram, quadrilateral. kite, quadrilateral. rectangle, parallelogram, quadrilateral. trapezoid, isosceles trapezoid, quadrilateral. square, rectangle, parallelogram, rhombus, quadrilateral. = ; = = = = 0. m = ; s = ; = M = ; = M =. f = ; g = ; FG = GH = HI = IF =. parallelogram. rectangle. kite. parallelogram Guided roblem olving -. a labeled figure, which shows an isosceles trapezoid. he nonparallel sides are congruent.. the measures of the angles and the lengths of the sides. m&g = c. c + (c - 0) = m& = m&g = 0, m& = m&f = 0. a - =. 0. = FG =, F =, G = = 0 = ( - )0. m& = m&g =, m& = m&f = ractice ; 0; 0. 0; 0; 0. ; ;. ; ; 0. ; 0;. ; ; 0; 0. ; ; Guided roblem olving -. the ratio of two different angle measures in a parallelogram. he consecutive angles are supplementar... the measures of the angles. he angles are supplementar angles, because the are consecutive.. + = 0. and. o; the lengths of the sides are irrelevant in this problem.. 0 and 0 ractice -. no. es. es. no. es. es. = ; =. = ; =. = ; = 0 0. = ; the figure is a because both pairs of opposite sides are congruent.. = 0; the figure is not a because one pair of opposite angles is not congruent.. = ; the figure is a because the congruent opposite sides are b the onverse of the lternate Interior ngles heorem.. Yes; the diagonals bisect each other.. o; the congruent opposite sides do not have to be.. o; the figure could be a trapezoid.. Yes; both pairs of opposite sides are congruent.. Yes; both pairs of opposite sides are b the converse of the lternate Interior ngles heorem.. o; onl one pair of opposite angles is congruent.. Yes; one pair of opposite sides is both congruent and. 0. o; onl one pair of opposite sides is congruent. Geometr ll-in-ne nswers Version

86 Geometr: ll-in-ne nswers Version (continued) Guided roblem olving -. a labeled figure, which shows a quadrilateral that appears to be a parallelogram. he consecutive angles are supplementar.. find values for and which make the quadrilateral a parallelogram. m& + m& = 0, so that & and & meet the requirements for same-side interior angles on the transversal of two parallel lines (heorems - and -).. & > &, b heorem = 0; + =. =, =. m& = m& = and m& = m& =, which matches the appearance of the figure.. ( + 0) + ( + ) = 0; es ractice - a. rhombus b. ; ; ; a. rectangle b. ; ; ; a. rectangle b. ; ; 0; a. rhombus b. ; 0; 0; a. rectangle b. 0; 0; 0; 0 a. rhombus b. ; ; ; 0. Yes; the parallelogram is a rhombus.. ossible; opposite angles are congruent in a parallelogram.. Impossible; if the diagonals are perpendicular, then the parallelogram should be a rhombus, but the sides are not of equal length. 0. = ; HJ = ; IK =. = ; HJ = ; IK =. = ; HJ = ; IK =. =-; HJ = ; IK = a. 0; 0; ; b. cm a. 0; 0; 0; 0 b. in. a. ; 0; 0; b. 0 m. possible. Impossible; because opposite angles are congruent and supplementar, for the figure to be a parallelogram the must measure 0, the figure therefore must be a rectangle. Guided roblem olving -. labeled figure, which shows a parallelogram. ne angle is a right angle, and two adjacent sides are congruent. lgebraic epressions are given for the lengths of three line segments.. diagonals. Find the values of and.. It is a square. heorem - and the fact that > impl that all four sides are congruent. heorems - and - plus the fact that m& = 0 impl that all four angles are right angles.. congruent; bisect. - + = ( - ) + ( + ); - = +. = ; =. It was not necessar to know >, but it was necessar to know m& = 0. he ke fact, which enables the use of heorem - in addition to heorem -, is that is a rectangle. It does not matter whether all four sides are congruent.. 0 ractice -. ;. ;. ;. ;. 0;. ;. =. = ; =. = 0. 0.; 0.. 0;. ;. 0; ;. 0; 0. 0; ;. =. =. = ; = Guided roblem olving -. Isosceles trapezoid with >. & > & and & > &. > is Given. > because opposite sides of a parallelogram are congruent (heorem -). > is from the ransitive ropert of ongruenc.. Isosceles; >; because base angles of an isosceles triangle are congruent.. & > & because corresponding angles on a transversal of two parallel lines are congruent.. & > & b the ransitive ropert of ongruenc.. & is a same-side interior angle with &, and & is a same-side interior angle with &.. his is not a problem, because for > there is a similar proof with a line segment drawn from to a point ling on.. he two drawn segments can be shown to be congruent, and then one has two congruent right triangles b the H heorem. & > & follows b and & > & because the are supplements of congruent angles. ractice -. (.a, b); a. (.a, b); "a b. (0.a, 0); a b. (0.a, b); "a b a b b b 0. - a. a. - a. (a, b); I(a, 0). (a, b); M(a, -b); (-a, -b). (a, b); I(a, 0). (0, b); (a, b); (a, b). (-a, b). (-b, 0) Guided roblem olving -. a rhombus with coordinates given for two vertices. rhombus is a parallelogram with four congruent sides.. the coordinates of the other two vertices. he are the diagonals.. he bisect each other.. W(-r, 0), Z(0, -t). o; neither heorem - nor an other theorem or result would appl.. lope of WX = 0 and slope of YZ is undefined. his confirms heorem -0, which sas that the diagonals of a rhombus are perpendicular. ractice - p p a. q b. = m + b; q = q (p) +b; b=q- q ; p p = +q- p q q c. = r + p d. = q (r + p) + p q - rp p ;= + p + q - rp q q q q ;= q + q; intersection rp r at (r + p, q + q) e. q f. (r, q) g. = m + b; r r q = (r) + b; b = q - r r ;= + q - q q q q r r h. = (r + p) + q - r ;= rp q q q + + q - r q q ; rp rp = q + q; intersection at (r + p, q + q) rp i. (r +p, q +q) a. (-a, 0) b. (-a, b) a b b c. (-, ) d. a a. (-a, 0) b. (-a, a) c. d. (a, -a) e.. he coordinates for are (0, b). he coordinates for are (a, 0). Given these coordinates, the lengths of and H can be determined: p ll-in-ne nswers Version Geometr

87 Geometr: ll-in-ne nswers Version (continued) = "(a 0) (0 b) = "a b ; H = "(0 a) (0 b) = "a b ; = H, so H. Guided roblem olving -. kite FG with = F with the midpoint of each side identified. kite is a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent.. he midpoints are the vertices of a rectangle.. (-b,c), G(0, 0). (b, a + c), M(b, c), (-b, c), K(-b, a + c). lope of K = slope of M = 0, slopes of K and M are undefined. pposite sides are parallel; it is a rectangle.. djacent sides are perpendicular.. right angles 0. nswers will var. ample: a =, b =, c = ields the points (-, ), (0, ), F(, ), G(0, 0) with midpoints at (-, ), (-, ), (, ), and (, ). onnecting these midpoints forms a rectangle.. onstruct F and G. lope of F = 0, so F is horizontal. lope of G is undefined, so G is vertical. : Graphic rganizer. Quadrilaterals. nswers ma var. ample: classifing quadrilaterals; properties of parallelograms; proving that a quadrilateral is a parallelogram; and special parallelograms. heck students work. : eading omprehension. Q >, Q >, Q, Q V. o, it cannot be proven that nqv > nu because with H M U the given information, onl one side and one angle of the two triangles can be proven to be congruent. nother side or angle is needed. If it were given that QUV is a parallelogram, then the proof could be made.. ll four sides are congruent.. Yes. ince G > G b the efleive ropert, nfg > nhg b.. b : eading/writing Math mbols. H, H, or H. G, FG, or G. G.. rhombus. rectangle. square. isosceles trapezoid : Visual Vocabular ractice/high-use dademic Words. solve. deduce. equivalent. indirect. equal. analsis. identif. convert. common : Visual Vocabular heck onsecutive angles: ngles of a polgon that share a common side. Kite: quadrilateral with two pairs of congruent adjacent sides and no opposite sides congruent. arallelogram: quadrilateral with two pairs of parallel sides. hombus: parallelogram with four congruent sides. rapezoid: quadrilateral with eactl one pair of parallel sides. F: Vocabular eview uzzle G Z I Q U M I G M I I U M G 0 H Y U 0 Geometr ll-in-ne nswers Version

88 Geometr: ll-in-ne nswers Version (continued) hapter ractice -. :. ft b 0 ft. ft b ft. ft b ft. ft b ft. ft b 0 ft. true. false. true 0. true. false. true. true. false. false :. :. :.. Guided roblem olving -. ratios.. the denominator,000,000 or,000,000.. ross-roduct ropert. 0.0,000,000, es.. ractice -. XYZ, with similarit ratio :. QM, with similarit ratio :. ot similar; corresponding sides are not proportional.. ot similar; corresponding angles are not congruent.. KM, with similarit ratio :. ot similar; corresponding sides are not proportional.. I.. J ft.. in... cm. 0 m Guided roblem olving -. equal....;...;.. no.. no..0;.. ince the quotients are not equal, the ratios are not equal, and the bills are not similar rectangles...0 ractice -. X XQ because vertical angles are. (Given). herefore X XQ b the M X XM ostulate.. ecause W = W = =, MX W b the heorem.. QM M QM M because vertical s are. hen, because M = M =, QM M b the heorem.. M (Given). ecause there are 0 in a triangle, m J = 0, and J. o MJ b the ostulate. X X. ecause X = X and X = X, X = X. X X because vertical angles are. herefore X X b the heorem.. ecause = = and XY = YZ = ZX, XY = YZ = ZX. hen XYZ b the heorem ft Guided roblem olving -. no; /. es; W,. It is a trapezoid.. he are congruent.. he are parallel.. he are congruent.. #Z and #WZ. ~ or ngle-ngle imilarit ostulate. o; there is onl one pair of congruent angles. 0. es; parallelogram, rhombus, rectangle, and square ractice -... ". ". 0". ". h.. 0. a. b. c. = ; = ". = "; = ".. = "; = ". = "; = "; z = ". = ; = "; z = ". " in. Guided roblem olving -. #, #, #.. ;... ;.. ". " 0. es. no (his would require the thagorean heorem.) ractice - JG.. H.. J. JI = ; =.. = ; =. = ; =... 0 Guided roblem olving -. parallel. ;. ;. 0. es. es. he sides would not be parallel.. he are similar triangles. : Graphic rganizer. imilarit. nswers ma var. ample: ratios and proportions; similar polgons; proving triangles similar; and similarit in right triangles. heck students work. : eading omprehension. H. similarit postulate he triangles have two similar angles.. :. :.. 0 ft. a H : eading/writing Math mbols. no. no. es. no. es. no. es,. es, Hpotenuse-eg heorem. es, or or 0. not possible : Visual Vocabular ractice. ngle-ngle imilarit ostulate. golden ratio. ide-ide-ide imilarit heorem. geometric mean. scale. ross-roduct ropert. golden rectangle. simplest radical form. ide-ngle-ide imilarit heorem ll-in-ne nswers Version Geometr

89 Geometr: ll-in-ne nswers Version (continued) : Vocabular heck imilarit ratio: he ratio of lengths of corresponding sides of similar polgons. ross-roduct ropert: he product of the etremes of a proportion is equal to the product of the means. atio: comparison of two quantities b division. Golden rectangle: rectangle that can be divided into a square and a rectangle that is similar to the original rectangle. cale: he ratio of an length in a scale drawing to the corresponding actual length. F: Vocabular eview... I.. J. K... G 0... M. H.. F hapter ractice -. ". ". ". ". ". ". in.. ft. cm 0. m. acute. right. obtuse. right. obtuse. acute Guided roblem olving -. the sum of the lengths of the sides. thagorean heorem. cm. cm cm. c = +.. cm. perimeter of rectangle = cm; es. nswers will var; eample: raw a cm cm grid, cop the given figure, measure the lengths with a ruler, add them together cm ractice - 0". = ; = ". a =.; b =.". c = ; " " d =. ". "... = ; = ". " 0. cm.. in.. 0. ft, ft. 0 in. 0". w = ; = ; " = ";z =. a = ; b =. p = "; q = ";r = ; s = " Guided roblem olving triangle. l. h.. Í or Í!.. or Í. ft. 0. min 0. es!. ft ractice -. tan = ; tan F =. tan = ; tan F =. tan = ; tan F = Guided roblem olving m& = m&x. 0. base: 0 cm, height: 0 cm es. ractice - "0. sin = ; cos =. sin = ; cos = ". sin = ; cos =. sin = ; cos = " ". sin = ; cos =. sin = ; cos = Guided roblem olving -. he sides are parallel.. sine. sin 0 w..0. es. cosine. cos.. nswers ma var. ample: cos 0 ; sin 0..,. ractice - a. angle of depression from the birds to the ship b. angle of elevation from the ship to the birds c. angle of depression from the ship to the submarine d. angle of elevation from the submarine to the ship a. angle of depression from the plane to the person b. angle of elevation from the person to the plane c. angle of depression from the person to the sailboat d. angle of elevation from the sailboat to the person.. ft.. ft.. ft.. d.. m.. m a. 0 cm 0 ft 0 cm b. ft Geometr ll-in-ne nswers Version

90 Geometr: ll-in-ne nswers Version (continued) Guided roblem olving -. &e =, &d =&. congruent. m&e = m&d. - = ( + ).... es., ractice -..0,.0. -., -..., mi/h;.º north of east.. m;.0º south of west.. m/sec;.º south of east. º north of east. 0º west of south. º west of north 0a., 0b. a., - b. a. -, - b.. ample: W. ample: W 0 Guided roblem olving -. heck students work.. 00 ; cos 0 ; 00 sin 0..; 0.., 0.., -0.., 0. ; due east. es 0. 00; due east : Graphic rganizer. ight riangles and rigonometr. nswers ma var. ample: the thagorean heorem; special right triangles; the tangent ratio; sine and cosine ratios; angles of elevation and depression; vectors. heck students work. : eading omprehension.. J.. J... b : eading/writing Math mbols. F. G..... H.. sin 0. #, #XYZ. m& <. tan Z : Visual Vocabular ractice triangle. inverse of tangent. congruent sides. tangent. thagorean heorem. hpotenuse triangle. thagorean triple. obtuse triangle : Vocabular heck btuse triangle: triangle with one angle whose measure is between 0 and 0. Isosceles triangle: triangle that has at least two congruent sides. Hpotenuse: he side opposite the right angle in a right triangle. ight triangle: triangle that contains one right angle. thagorean triple: set of three nonzero whole numbers a, b, and c that satisf the equation a + b = c. ll-in-ne nswers Version Geometr

91 hapter ractice -. o; the triangles are not the same size.. Yes; the heagons are the same shape and size.. Yes; the ovals are the same shape and size. a. and F b. and, and, and, and a. M and b. and, and, and. (, ) ( -, - ). (, ) ( -, - ). (, ) ( -, - ). (, ) ( +, - ) 0. (, ) ( -, + ). (, ) ( +, + ). W (-, ), X (-, ), Y (, ), Z (, ). J (-, 0), K (-, ), (-, -). M (, -), (, -), (, -), Q (, -). (, ) ( +., +.). (, ) ( +, - ). (, ) (,). (, ) ( +, + ) a. (-, -) b. (0, ), (-, ), Q (, ) Guided roblem olving -. the four vertices of a preimage and one of the vertices of the image. Graph the image and preimage.. (, ) and (0, 0). =, =, + a = 0, + b = 0. a =-; b =-; (, ) ( -, - ). (-, ), (, ), (-, ).. es. (, ) ( +, - ), (, ), (, ), (, 0) ractice -. (-, -). (-, -). (-, -). (, -). (, -). (, -) a. K J I M M M M F F F F Geometr: ll-in-ne nswers Version (continued) Geometr ll-in-ne nswers Version F: Vocabular eview uzzle J H I Z H H Y M Y K W I Y F H V M M G G I U X Z I U I G V U Z K X I U I U F U F U Q X F V X G I G I M F M U M G Y Y M Q Y J Z V M H G M I M M F I J H Q M X M I J I V V K G I Y H G

92 Geometr: ll-in-ne nswers Version (continued) b. K J I. 0, a. Z Y b W X Y Z X W. (-, ). (-, 0). (0, -) Guided roblem olving -. a point at the origin, and two reflection lines. reflection is an isometr in which a figure and its image have opposite orientations.. the image after two successive reflections... = (0,0) = (0,0) = (0,0) "(0, ) '(0,) '(0,). (0, - ). Yes. he -coordinate remains 0 throughout.. (0, 0) and (0, ) ll-in-ne nswers Version Geometr

93 Geometr: ll-in-ne nswers Version (continued) ractice -. I. I. I. GH. G.. U. slope of Q'. ' Q M M Q Q ' U Q Guided roblem olving -. he coordinates of point, and three rotation transformations. It is assumed that the rotations are counterclockwise.. parallelogram, rhombus, square M M Q Q Q M ' Q Q Q' '.. slope of ; slope of ; slope of ; the slopes of perpendicular line segments are negative reciprocals.. square. es. (, ), (, -), (-, -) ractice -. he helmet has reflectional smmetr.. he teapot has reflectional smmetr.. he hat has both rotational and reflectional smmetr.. he hairbrush has reflectional smmetr.... his figure has no lines of smmetr... line smmetr line smmetr and rotational smmetr Geometr ll-in-ne nswers Version

94 Geometr: ll-in-ne nswers Version (continued) X X.. line smmetr. 0. line smmetr and 0 rotational smmetr line smmetr and rotational smmetr 0 rotational smmetr line smmetr line smmetr and rotational smmetr 0 rotational smmetr K H X Guided roblem olving -. the coordinates of one verte of a figure that is smmetric about the -ais. ine smmetr is the tpe of smmetr for which there is a reflection that maps a figure onto itself.. the coordinates of another verte of the figure. images (and preimages). reflection across the -ais. (-, ). Yes. (-, ) ractice -. (-, -), M (-, 0), (, -), (0, -). (-0, -0), M (-, 0), (0, -), (0, -). (-, -), M (-, 0), (, -), (0, -).... es. no. no (-, -), Q (-, 0), (0, -). (-, ), Q (,- ), (,). (-, ), Q (, ), (-, ). (-, ), Q (-, 0), (0, ) Guided roblem olving -. description of a square projected onto a screen b an overhead projector, including the square s area and the scale factor in relation to the square on the transparenc.. he scale factor of a dilation is the number that describes the size change from an original figure to its image.. the area of the square on the transparenc. smaller; he scale factor >, so the dilation is an enlargement.. ; ll-in-ne nswers Version Geometr

95 Geometr: ll-in-ne nswers Version (continued). ;eing as high and as wide, the square on the tranparenc has times the area.. ft. he shape does not matter. egardless of the shape, the figure is being enlarged b a factor of in two directions, so that the screen image has = as large an area as the figure on the transparenc.. 0 ft. ractice -. I. II. III. IV.. I. II. III. IV J 0 J m m m J m m reflection. rotation. glide reflection. translation Geometr ll-in-ne nswers Version

96 Geometr: ll-in-ne nswers Version (continued) Guided roblem olving -. assorted triangles and a set of coordinate aes. a transformation. the transformation that maps one triangle onto another. he are congruent, which confirms that the could be related b one of the isometries listed. orresponding vertices: and, and Q, and M.. counterclockwise; clockwise. he triangles have opposite orientations, so the isometr must be a reflection or glide reflection.. o; there is no reflection line that will work.. a glide reflection consisting of a translation (, ) ( +, ) followed b a reflection across line = 0 (the -ais). Yes, both are horizontal.. 0º rotation with center Q, 0 ractice translational smmetr line smmetr, rotational smmetr, translational smmetr, glide reflectional smmetr translational smmetr line smmetr, rotational smmetr, translational smmetr, glide reflectional smmetr line smmetr, rotational smmetr, translational smmetr, glide reflectional smmetr.... amples: es. es. no. es. no. no Guided roblem olving -. two different-sized equilateral triangles and a trapezoid. tessellation is a repeating pattern of figures that completel covers a plane, without gaps or overlaps.. heorem - sas that ever quadrilateral tessellates. If the three polgons can be joined to form a quadrilateral (using each polgon at least once), that quadrilateral can be the basis of a tessellation... ossible answer:. heck students work.. nswers ma var. ample:. rotational smmetr, translational smmetr : Graphic rganizer. ransformations. nswers ma var. ample: reflections; translations; compositions of reflections; and smmetr. heck students work. ll-in-ne nswers Version Geometr

97 Geometr: ll-in-ne nswers Version (continued) : eading omprehension..,,, and 0. he center of the count is the center of the square mile bounded b roads,, K and... 0 miles. a : eading/writing Math mbols. reflection; opposite. rotation; same. glide reflection; opposite. translation; same. rotation; same. translation; same : Visual Vocabular ractice/high-use cademic Words. theorem. smbols. reduce. application. simplif. eplain. characteristic. table. investigate hapter 0 M 0 I Y U M M M G I Y F Y : Vocabular heck ransformation: change in the position, size, or shape of a geometric figure. reimage: he given figure before a transformation is applied. Image: he resulting figure after a transformation is applied. Isometr: transformation in which an original figure and its image are congruent. omposition: transformation in which a second transformation is performed on the image of the first transformation. F: Vocabular eview uzzle Y H I H I I I I I U U I I F ractice M V I X Guided roblem olving 0-.. heck students work; (, ). a right angle bh.. å.... a square b 0 Geometr ll-in-ne nswers Version

98 Geometr: ll-in-ne nswers Version (continued) ractice 0-. cm. in... ft. 0 m. 00 in.. ft. 0 cm.. in.. ft 0.. square units. " square units. " square units. cm.. in..,00 m Guided roblem olving 0-.. ; ;.. no h(b b ). es. ractice 0- " ". = ; =.". a = 0; c = ; d =. p = ";q = ; = 0. m = ; m = ; m = 0; m = 0. m = 0; m = 0; m = 0. m = 0; m = 0; m 0 = 0. "..". " 0. 0 in.. in.. in. Guided roblem olving he angles are all congruent and the sides are all congruent bisects ; ; ractice 0-. : ; :. : ; :. : ; :. :. :. :. :. in. 00. cm 0. in.. :. :. :. 0 ft Guided roblem olving 0-. scale. heck students work.. nswers will var. et a be the side opposite the 0 angle, b be the side opposite the 0 angle, and c be the side corresponding to 00 d. hen 00 d a < 0 mm, b < 0 mm and c < 0 mm.. 0 mm d mm. < 0 mm. < mm. < 0 mm es.,000 ractice 0-.. cm. ft. 0. mm.. m.. in... cm.. in.. cm.. m mi.. km.. in... mm..0 in... cm.. ft. 0. m.. m..0 ft 0.. ft. m.. ft Guided roblem olving 0-. subtract. 0. ap tan 0.. es 0. ractice 0-. p. p..p. amples: F, F 0 0. amples: F, F. amples: F,. amples: 0 0 F,.. F p in.. p cm. p m Guided roblem olving 0-. ;. ; ;. circumference; ractice 0- p. p... p. p -.. p or. p p. p or. -. p 0. p. p. p p. p. p. p. p Guided roblem olving 0-. m a piece from 0? pr the outer ring. es. a piece from the top tier ractice 0-..%. 0.%..% %..0%..%. 0%. 0%. 0%. % Guided roblem olving 0-. outcomes. event. F. FG. G. ; heck students work.. or about %. or about %; es. or 0% 0: Graphic rganizer. rea. nswers ma var. ample: areas of parallelograms and triangles; areas of trapezoids, rhombuses, and kites; areas of regular polgons; perimeters and areas of similar figures; trigonometr and area; circles and arcs; areas of circles and sectors; geometric probabilit. heck students work. 0: eading omprehension.. J.. G.. I... H 0. F. a + b = c 0 å å. m 0. M 0 0. ". #JK. XY > Q. ~. =. b 0: eading/writing Math mbols. 0 ft. ft. bundles. squares. $.00. $.00. minutes.. hours. $ 0. $ ll-in-ne nswers Version Geometr

99 Geometr: ll-in-ne nswers Version (continued) 0: Visual Vocabular ractice. area of a triangle. radius. altitude of a parallelogram. area of a trapezoid. area of a kite or rhombus. similarit ratio. apothem. perimeter (of an octagon). height of a triangle 0: Vocabular heck emicircle: Half a circle. djacent arcs: rcs on the same circle with eactl one point in common. pothem: he distance from the center to the side of a regular polgon. ircumference: he distance around a circle. oncentric circles: ircles that lie on the same plane and have the same center. 0F: Vocabular eview uzzle. adjacent arcs. semicircle. center. arc length. base. circle. central angle. congruent arcs. diameter 0. major arc. concentric circles. minor arc. thagorean triple. radius. apothem. sector. height. circumference Highness, there is no roal road to geometr. hapter ractice triangle. rectangle 0. rectangle. rectangle. rectangle Guided roblem olving -. a two-dimensional network. Verif uler s Formula for the network shown... ;. + = + ; his is true.. F becomes, V stas at, becomes. uler s Formula becomes + = +, which is still true.. + = + ; his is true. ractice -.. cm.. ft.. cm.. m..0 ft.. cm a. 0 m b. 0 m a. in. b. in. a. mm b. mm 0a. cm 0b. 0 cm a. 00 m b. 00 m a. 000 ft b. 0 ft. p m..p cm. 0p ft Guided roblem olving -. a picture of a bo, with dimensions. the amount of cardboard the bo is made of, in square inches. front and a back, each in. in.; a top and a bottom, each in. in.; and one narrow side, in. in.. Front and back: each 0 in. ; top and bottom: each in. ; one narrow side: in.. in.. in. is about half a square foot ( ft = in. ), which is reasonable. he answer is an approimation because the solution did not consider the overlap of cardboard at the glued joints, nor does it factor in the trapezoid-shaped cutout in the front and back sides.. in. ractice -. m. 0 cm. ft. p cm. 00p m. 00p in.. 0. m. 0.0 cm.. ft 0.. m.. cm.. ft Guided roblem olving -. a solid consisting of a clinder and a cone. the surface area of the solid. he top of the solid is a circular base of the clinder, = pr. he side of the solid is the lateral area of the clinder, = prh. he bottom of the solid is the lateral area of the cone, = pr/.. r = ft; h = ft; / " ft. op: =. ft ; side: =. ft ; bottom: 0. ft. ft. bout 0., or a little less than half. his seems reasonable.. ft ractice -..0 m.,. cm.. in... in... cm. 0. cm. 0 in.. ft. m 0. ft. cm. in.. ft. in.. m Guided roblem olving -. dimensions for a laer of topsoil, and two options for buing the soil. which purchasing option is cheaper. ft. 00 ft. bags; $ ft < d. $ (or slightl less if the topsoil volume is not rounded up to the net whole number). buing in bulk. he two costs are quite close. It makes sense that buing in bulk would be slightl cheaper for a good-sized backard. 0. uing topsoil b the bag would be less epensive because it would cost $ and in bulk would cost at least $.. ractice -., cm. 00 in.. 0,0 in.. 00 d. 0 m. cm.. cm.. in... m 0.. in m.. ft. 0.. Geometr ll-in-ne nswers Version

100 Geometr: ll-in-ne nswers Version (continued) Guided roblem olving -. a description of a plumb bob, with dimensions. the plumb bob s volume. equilateral; cm. a = " cm; = " cm. rism: V = " cm ; pramid: V = " cm ; plum bob: V < cm. ; this seems reasonable. cm ractice -..0 in..,,. m.. cm.. m. 0. ft.. m.. mi.,0. cm.. m 0.. cm.,,. cm. 0,0. d. ft. m. cm. cm. cm. cm. cm Guided roblem olving -. the diameter of a hailstone, along with its densit compared to normal ice. the hailstone s weight. V.. in... in. pr.. lb. he densit of normal ice. his was given for comparison purposes onl oz ractice -. :. :. es; :. es; :. not similar. es; :. cm. in.. ft 0. ft. 0 m. 00 cm.,000 lb. lb Guided roblem olving -. two different sizes for an image on a balloon, and the volume of air in the balloon for one of those sizes. the volume of air for the other size. he two sizes of the clown face have a ratio of :, or :.. :. 0 in. = in.. Yes, the two clown face heights are lengths, measured in inches (not in. or in. ).. cm : Graphic rganizer. urface rea and Volume. nswers ma var. ample: space figures and nets; space figures and drawing; surface areas; and volumes.. heck students work. : eading omprehension. rea = base times height; area of a triangle. slope = the difference of the -coordinates divided b the difference of the -coordinates; slope of the line between points. a squared plus b squared equals c squared, the thagorean heorem; the lengths of the sides of a right triangle. area = pi times radius squared; area of a circle. volume = pi times radius squared times height; volume of a clinder. the sum of the -coordinates of two points divided b, and the sum of the -coordinates of two points divided b ; midpoint of a line segment with endpoints (, ) and (, ). circumference = two times pi times the radius; circumference of a circle. volume = one-third pi times radius squared times height; volume of a cone. distance equals the square root of the quantit of the sum of the square of the difference of the -coordinates, squared, plus the difference of the -coordinates, squared; the distance between two points (, ) and (, ) 0. area = one-half the height times the sum of base one plus base two; area of a trapezoid. a : eading/writing Math mbols.. in. ft : Visual Vocabular ractice. prism. cross section. sphere. clinder. cone. pramid. polhedron. altitude. base : Vocabular heck Face: surface of a polhedron. dge: n intersection of two faces of a polhedron. ltitude: For a prism or clinder, a perpendicular segment that joins the planes of the bases. urface area: For a prism, clinder, pramid, or cone, the sum of the lateral area and the areas of the bases. Volume: measure of the space a figure occupies. F: Vocabular eview. polhedron. edge. verte. cross section. prism. altitude. right prism. slant height. cone 0. volume. sphere. hemispheres. similar figures. hpotenuse hapter ractice ".. ". "0. " 0. circumscribed. inscribed. circumscribed. cm. in.. ft Guided roblem olving -. he area of one is twice the area of the other.. inscribed; circumscribed... r. r. "r. r. r = (r ): ne area is double the other area. 0. es. he same: he area of one circle is twice the area of the other circle. r ll-in-ne nswers Version Geometr

101 Geometr: ll-in-ne nswers Version (continued) ractice r = ; m.. r = ";m. # 0. r = ; m Q ; U. J; K. onstruct radii and. F = because a diameter perpendicular to a chord bisects it, and chords and are congruent (given). hen, b H, F, and b, = F.. ecause congruent arcs have congruent chords, = =. hen, because an equilateral triangle is equiangular, m = m = m. Guided roblem olving -. perpendicular. bisects. ;. right. radius. thagorean heorem.. es. ; ractice -. and ; and. and. and.. = ; = 0; z =. = 0; = = 0; = 0; z = 0 0. = 0; = 00; z = 0. = 0 ; = ; z =. = 0; = 0; z = 0 a. m = 0 b. m = c. m = 0 d. m = a. m = 0 b. m = 0 c. m = 0 d. m = 00 Guided roblem olving a diameter. heck students work.. heck students work.. heck students work.. heck students work.. 0. heck students work. ractice = ; = 0;. 0; = 0. = ; = ; z =. = 0; =. = 0; = 0; z = 0 0. = ; =. = ; = ; z =. = 0; = Guided roblem olving -. tangent ; ; es; he sum of the measures of the arcs should be 0.. pposite angles are not supplementar. ractice -. (0, 0), r =. (, ), r =. (-, -), r =. (-, ), r = ". + =. ( - ) +( - ) =. ( - ) +( - ) =. ( + ) + ( - ) =. ( + ) + ( + ) = 0. ( + ) + ( - ) =. + =. ( + ) + ( - ) =. + ( - ) =. ( - ) + ( + ) =. + ( + 0) = 00. ( + ) + ( + ) =. (0, ).. 0. (, ) (, ) (, ) (0, 0) (, ). + =. ( - ) + ( - ) =. ( + ) + ( + ) =. ( - ) + ( + ) = 0. ( + ) + ( + ) =. ( - ) + ( - ) = (, 0) (, ) Geometr ll-in-ne nswers Version

102 Geometr: ll-in-ne nswers Version (continued) Guided roblem olving -. (, );.. he are perpendicular ; es. -intercept: a = ; -intercept: b = ractice -... U mm mm 0. cm.... X 0. in. 0. in.... a third line parallel to the two given lines and midwa between them. a sphere whose center is the given point and whose radius is the given distance 0. the points within, but not on, a circle of radius in. centered at the given point.. cm. cm Q Y in. Q 0. in. 0. in. F H G J 0. in. 0. in. Guided roblem olving -. a line. heck students work.. he are perpendicular the midpoint of Q. (, ). -. = - 0. (, 0); es. : Graphic rganizer. ircles. nswers ma var. ample: tangent lines; chords and arcs; inscribed angles; and angle measures and segment lengths. heck students work. : eading omprehension nswers ma var. ample answers:.,. F,,. H. H F,,. F,,.,. /. / 0.. /H. /G. /H. /. H. a : * ) eading/writing * ) Math mbols ' F.. is tangent to (G.. WX > ZX. ' XY. HM = : Visual Vocabular ractice. tangent to a circle. standard form of an equation of a circle. secant. chord. circumscribed about. intercepted arc. inscribed in. inscribed angle. locus : Vocabular heck Inscribed in: circle inside a polgon with the sides of the polgon tangent to the circle. hord: segment whose endpoints are on a circle. oint of tangenc: he single point of intersection of a tangent line and a circle. angent to a circle: line, segment, or ra in the plane of a circle that intersects the circle in eactl one point. ircumscribed about: circle outside a polgon with the vertices of the polgon on the circle. F: Vocabular eview.. F. G..... M. I 0... H. K. J ll-in-ne nswers Version Geometr

103 Geometr: ll-in-ne nswers Version ame lass ate ame lass ate esson - esson bjective Use inductive reasoning to make conjectures Vocabular. Inductive reasoning is 00 trand: Geometr opic: Mathematical easoning atterns and Inductive easoning reasoning based on patterns ou observe. Finding a ountereample Find a countereample for the conjecture. ince, the sum of the squares of two consecutive numbers is the square of the net consecutive number. egin with. he net consecutive number is. he sum of the squares of these two consecutive numbers is. he conjecture is that is the square of the net consecutive number. he square of the net consecutive number is ( ). conjecture is countereample is amples. FindingandUsingaattern Find a pattern for the sequence. Use the pattern to find the net two terms in the sequence.,,,,... ach term is half the preceding term. he net two terms are and. Using Inductive easoning Make a conjecture about the sum of the cubes of the first counting numbers. Find the first few sums. otice that each sum is a perfect square and that the perfect squares form a pattern. ( ) ( ) 00 0 ( ) ( ) he sum of the first two cubes equals the square of the sum of the first two counting numbers. he sum of the first three cubes equals the square of the sum of the first three counting numbers. his pattern continues for the fourth and fifth rows. o a conjecture might be that the sum of the cubes of the first counting numbers equals the square of the sum of the first counting numbers, or (... ). Geometr esson - a conclusion ou reach using inductive reasoning. an eample for which the conjecture is incorrect. ail otetaking Guide ame lass ate. Make a conjecture about the sum of the first odd numbers. Use our calculator to verif our conjecture. he sum of the first odd numbers is, or.. Finding a ountereample Find a countereample for the conjecture. ome products of and other numbers are shown in the table. herefore, the product of and an positive integer ends in. ince 0, not all products of end in. (ut the all either end in or 0.). uppose the price of two-da shipping was $.00 in 000, $.00 in 00, and $.00 in 00. Make a conjecture about the price in 00. ach ear the price increased b $.00. possible conjecture is that the price in 00 will increase b $.00. If so, the price in 00 would be $.00 $.00 $.00. o, since and are not the same, this conjecture is false. ppling onjectures to usiness he price of overnight shipping was $.00 in 000, $.0 in 00, and $.00 in 00. Make a conjecture about the price in 00. Write the data in a table. Find a pattern. ach ear the price increased b $.0 the price in 00 will increase b $.0 be $.00 $.0 $.0. Quick heck.. Find the net two terms in each sequence. a.,,,,,,,,,.... possible conjecture is that. If so, the price in 00 would b. Monda, uesda, Wednesda, hursda, Frida,... c $.00 $.0 $.00 nswers ma var. ample:,,... ail otetaking Guide Geometr esson - ame lass ate esson - esson bjectives Make isometric and orthographic drawings raw nets for three-dimensional figures Vocabular. n isometric drawing of a three-dimensional object shows a corner view of the figure drawn on isometric dot paper. n orthographic drawing is the top view, front view, and right-side view of a three-dimensional figure. foundation drawing shows the base of a structure and the height of each part. net is a two-dimensional pattern ou can fold to form a three-dimensional figure. amples. Isometric rawing Make an isometric drawing of the cube structure at right. n isometric drawing shows three sides of a figure from a corner view. Hidden edges are not shown, so all edges are solid lines. 00 trand: Geometr opic: imension and hape rawings, ets, and ther Models Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

104 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate rthographic rawing Make an orthographic drawing of the isometric drawing at right. Identifing olids from ets Is the pattern a net for a cube? If so, name two letters that will be opposite faces. rthographic drawings flatten the depth of a figure. n orthographic drawing shows three views. ecause no edge of the isometric drawing is hidden in the top, front, and right views, all lines are solid. Front ight he pattern is a net because ou can fold it to form a cube. Fold squares and up to form the back and front of the cube. Fold up to form a side. Fold over to form the top. Fold F down to form another side. F fter the net is folded, the following pairs of letters are on opposite faces: and are the back and front faces. and are the bottom and top faces. Front op ight Foundation rawing Make a foundation drawing for the isometric drawing. o make a foundation drawing, use the top view of the orthographic drawing. ecause the top view is formed from squares, show squares in the foundation drawing. Identif the square that represents the tallest part. Write the number in the back square to indicate that the back section is cubes high. Write the number in each of the two front squares to indicate that each front section is cube high. Front op ight Geometr esson - Front ight ail otetaking Guide ame lass ate. Make an orthographic drawing from this isometric drawing. Front op ight. a. How man cubes would ou use to make the structure in ample? b. ritical hinking Which drawing did ou use to answer part (a), the foundation drawing or the isometric drawing? plain. nswers ma var. ample: the foundation drawing; ou can just add the three numbers.. ketch the three-dimensional figure that corresponds to the net. Front ight and F are the right and left side faces. rawing a et raw a net for the figure with a square base and four isosceles triangle faces. abel the net with its dimensions. hink of the sides of the square base as hinges, and unfold the figure at these edges to form a net. he base of each of the four isosceles triangle faces is a side of the square. Write in the known dimensions. cm 0 cm Quick heck.. Make an isometric drawing of the cube structure below. ail otetaking Guide Geometr esson - ame lass ate. he drawing shows one possible net for the Graham rackers bo. cm cm 0 cm GHM K cm cm GHM K raw a different net for this bo. how the dimensions in our diagram. nswers ma var. ample: cm cm 0 cm 0 cm 0 cm cm Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

105 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Understand basic terms of geometr Understand basic postulates of geometr oints, ines, and lanes 00 trand: Geometr opic: imension and hape plane is a flat surface that has no thickness. wo points or lines are coplanar if the lie on the same plane. postulate or aiom is an accepted statement of fact. lane Vocabular and Ke oncepts. 0 ostulate - hrough an two points there is eactl one line. ostulate - Geometr esson - ine t is the onl line that passes through points and. If two lines intersect, then the intersect in eactl one point. ostulate - and intersect at. If two planes intersect, then the intersect in eactl one line. W ostulate - t * ) * ) lane and plane W intersect in. hrough an three noncollinear points there is eactl one plane. point is a location. pace is the set of all points. line is a series of points that etends in two opposite directions without end. ollinear points are points that lie on the same line. * ) ail otetaking Guide ame lass ate Using ostulate - hade the plane that contains X, Y, and Z. oints X, Y, and Z are the vertices of one of the four triangular faces of the pramid. o shade the plane, shade the interior of the triangle formed b X, Y, and Z. Quick heck.. Use the figure in ample. a. re points W, Y, and X collinear? no b. ame line m in three different was. * ) * ) * ) nswers ma var. ample: ZW, WY, YZ. c. ritical hinking Wh do ou think * ) arrowheads are used when drawing a line or naming a line such as ZW? rrowheads are used to show that the line etends in opposite directions without end.. ist three different names for plane Z. nswers ma var. ample: HF, HFG, FGH.. ame two planes that intersect in. F and F. a. hade plane VWX. b. ame a point that is coplanar Z with points V, W, and X. Y Y X * F ) V Y Z W H Z H F t F G X G amples. Identifing ollinear oints In the figure at right, X name three points that are collinear and three points that are not collinear. oints Y, Z, and W lie on a line, so the are collinear. n other set of three points in the figure do not lie on a line, so no other set of three points is collinear. For eample, X, Y, m Z Y W and Z form a triangle and are not collinear. aming a lane ame the plane shown in two different was. You can name a plane using an three or more points on that plane that are not collinear. ome possible names for the plane shown are the following: plane, plane U, plane U, plane U, and plane U. Finding the Intersection of wo lanes Use the diagram at right. What is the intersection of plane HG and plane? s ou look at the cube, the front face is on plane F, the back face is on plane HG, and the left face is on plane. he back and left faces of the cube intersect at H. lanes HG and intersect * verticall at H ). ail otetaking Guide Geometr esson - ame lass ate esson - esson bjectives Identif segments and ras ecognize parallel lines Vocabular. segment is the part of a line consisting of two egment endpoints and all points between them. ndpoint ndpoint ra is the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint. pposite ras are two collinear ras with the same endpoint. arallel lines are coplanar lines that do not intersect. kew lines are noncoplanar; therefore, the are not parallel and do not intersect. arallel planes are planes that do not intersect. J G H G F H I 00 trand: Geometr opic: elationships mong Geometric Figures is parallel and G are lane is to F. skew lines. H to plane GHIJ. F U G egments, as, arallel ines and lanes YX ) Q ) and ) Q are opposite ras. parallel a YX X Y ndpoint V W Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

106 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. aming egments and as ame the segments and ras in the figure. he labeled points in the figure are,, and.. Use the diagram in ample. a. ame all labeled segments that are parallel to GF. H,, segment is a part of a line consisting of two endpoints and all points between them. segment is named b its two endpoints. o the segments are (or ) and (or ). ra is a part of a line consisting of one endpoint and all the points of the line on one side of that endpoint. ra is named b its endpoint first, followed b an other point on the ra. o the ras are ) and ). Identifing arallel and kew egments Use the figure at right. ame all segments that are parallel to. ame all segments that are skew to. arallel segments lie in the same plane, and the lines that contain them H do not intersect. he three segments in the figure that are parallel to are F, G and H. kew segments are segments that do not lie in the same plane. he four segments in the figure that do not lie in the same plane as are,, FG and GH. Identifing arallel lanes Identif a pair of parallel planes in our classroom. lanes are parallel if the do not intersect. If the walls of our classroom are vertical, opposite walls are parts of parallel planes. If the ceiling and floor of the classroom are level, the are parts of parallel planes. Quick heck. ) ). ritical hinking Use the figure in ample. and form a line. re the opposite ras? plain. o; the do not have the same endpoint. Geometr esson - F G ail otetaking Guide ame lass ate esson - esson bjectives Find the lengths of segments Vocabular and Ke oncepts. ostulate -: uler ostulate he points of a line can be put into one-to-one correspondence with the real numbers so that the distance between an two points is the absolute value of the difference of the corresponding numbers. ostulate -: egment ddition ostulate If three points,, and are collinear and is between and, then. coordinate is a point s distance and direction from zero on a number line. the length of u a b u a Q a coordinate of coordinate of ongruent ( ) segments are segments with the same length. cm cm midpoint is a point that divides a segment into two congruent segments. 00 trand: Measurement opic: Measuring hsical ttributes Measuring egments b midpoint b. ame all labeled segments that are skew to GF.,,, H c. ame another pair of parallel segments and another pair of skew segments. nswers ma var. ample: G, F; F, H. Use the diagram to the right. a. ame three pairs of parallel planes. W QVU, U QVW, Q WVU * Q ) b. ame a line that is parallel to. * ) V c. ame a line that is parallel to plane QUV. * ) nswers ma var. ample: ail otetaking Guide Geometr esson - ame lass ate amples. omparing egment engths Find and. re and congruent? 0 = - - ( ) «= -«= = - «=»» = so and are not congruent.. Using the egment ddition ostulate If, find the value of. hen find and. Use the egment ddition ostulate (ostulate -) to write an equation. egment ddition ostulate ( ) ( ) ubstitute. implif the left side. ubtract from each side. ivide each side b. ( ) 0 ubstitute for. ( ) 0 and, which checks because the sum of the segment lengths equals. W U Q V Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

107 0 0 0 Geometr: ll-in-ne nswers Version (continued) ame lass ate Using the Midpoint M is the midpoint of. Find M, M, and. M Use the definition of midpoint to write an equation. M M efinition of midpoint ubstitute. dd to each side. ame lass ate. G 00. Find the value of. hen find F and FG F G, F 0; FG 0 ubtract from each side. ivide each side b. M ( ) ubstitute for. M ( ) M M egment ddition ostulate M and M are each, which is half of, the length of. Quick heck.. Find. Find, different from, such that and are congruent. 0 ( ) We also want. hat means that must lie units on the other side of from. o must lie at. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Find the measures of angles Identif special angle pairs Vocabular and Ke oncepts. ostulate ) -: rotractor ) ostulate et and be opposite ras in a plane.,, and all the ras with * ) endpoint that can be drawn on one side of ) ) can be paired with the real numbers ) from 0 to 0 so that ) a. is ) paired with 0 and ) is paired with 0. b. If is paired with and is paired with, then ml u u ostulate -: ngle ddition ostulate If point is in the interior of, then m m m If is a straight angle, then m m trand: Measurement opic: Measuring hsical ttributes Measuring ngles. Z is the midpoint of XY, and XY. Find XZ.. ail otetaking Guide Geometr esson - ame lass ate n angle ( ) is formed b two ras with the same endpoint. he ras are the sides of the angle and the endpoint is the verte of the angle. n acute angle has a measurement between 0 and 0. right angle has a measurement of eactl 0. n obtuse angle has a measurement between 0 and 0. straight angle has a measurement of eactl 0. ongruent angles are two angles with the same measure. amples. aming ngles ame the angle at right in four was. he name can be the number between the sides of the angle: l. he name can be the verte of the angle: lg. Finall, the name can be a point on one side, the verte, and a point on the other side of the angle: lg or lg. G acute angle 0,, right angle 0 Measuring and lassifing ngles Find the measure of each /. lassif each as acute, right, obtuse, or straight. a. b obtuse angle 0,, Q Q straight angle , acute 0, obtuse 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

108 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Using the ngle ddition ostulate uppose that m and m. Find m. Use the ngle ddition ostulate (ostulate -) to solve. Quick heck.. a. ame two other was. m m m ngle ddition ostulate l, l m ubstitute for m and for m. m ubtract from each side. b. ritical hinking Would it be correct to name an of the angles? plain. Identifing ngle airs In the diagram identif pairs of numbered angles that are related as follows: a. complementar and b. supplementar and ; and c. vertical angles and Making onclusions From a iagram an ou make each conclusion from the diagram? a. Yes; the markings show the are congruent. b. and are supplementar o; there are no markings. c. m( ) m( ) 0 Yes; ou can conclude that the angles are supplementar from the diagram. d. Yes; the markings show the are congruent. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Use a compass and a straightedge to construct congruent segments and congruent angles Use a compass and a straightedge to bisect segments and angles Vocabular. onstruction is using a straightedge and a compass to draw a geometric figure. straightedge is a ruler with no markings on it. compass is a geometric tool used to draw circles and parts of circles called arcs. erpendicular lines are two lines that intersect to form right angles. perpendicular bisector of a segment is a line, segment, or ra that is perpendicular to the segment at its midpoint, thereb bisecting the segment into two congruent segments. n angle bisector is a ra that divides an angle into two congruent coplanar angles. amples. onstructing ongruent egments onstruct W congruent K to KM. tep raw a ra with endpoint. tep pen the compass the length of KM. tep With the same compass setting, put the compass point on point. raw an arc that intersects the ra. abel the point of intersection W. KM W 00 trand: Geometr opic: elationships mong Geometric Figures asic onstructions J K W M o, angles have for a verte, so ou need more information in the name to distinguish them from one another.. Find the measure of the angle. lassif it as acute, right, obtuse, or straight. 0; acute. If m G, find m GF.. ame an angle or angles in the diagram supplementar to each of the following: a. b. a. b. or F. an ou make each conclusion from the information in the diagram? plain. a. b. and are supplementar c. m( ) m( ) 0 a. Yes; the markings show the are congruent. b. Yes; ou can conclude the angles are adjacent and supplementar from the diagram. c. Yes; ou can conclude that the angles are supplementar from the diagram and their sum is 0 b the ngle ddition ostulate. ail otetaking Guide Geometr esson - ame lass ate onstructing ongruent ngles onstruct Y so that Y G. tep raw a ra with endpoint Y. Y tep With the compass point on point G, draw an arc that intersects both sides of G. abel the points of intersection and F. tep With the same compass setting, put the compass point on point Y. raw an arc that intersects the ra. abel the point of intersection Z. tep pen the compass to the length F. Keeping the same compass setting, put the compass point on Z. raw an arc that intersects with the arc ou drew in tep. abel the point of intersection X. ) tep raw YX to complete Y. G Y Y G G F X X Z Z Y G Y Z Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

109 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate onstructing the erpendicular isector Given:.* ) * ) onstruct: XY so that XY at the midpoint M of. Quick heck.. Use a straightedge to draw XY. hen construct so that XY. tep ut the compass point on point and draw a long arc. e sure that the opening is greater than. X Y tep With the same compass setting, put the compass point on point and draw another long arc. abel the points where the two arcs intersect as X and Y. tep raw. he point of intersection of and is M, the midpoint of. * XY ) * XY ) at the midpoint of, so is the perpendicular bisector of. Finding ngle Measures bisects W so that m W and m W. Find m W. W W m m efinition of angle bisector ubstitute m W and for for m W. ubtract from each side. ivide each side b. m W ubstitute for. m W ( ) W W ngle ddition m W m m ostulate m W ubstitute for m W and for m W. Geometr esson - Ke oncepts. * W ) * XY ) * XY ) Formula: he istance Formula he distance d between two points (, ) and (, ) is Formula: he Midpoint Formula he coordinates of the midpoint M of with endpoints (, ) and (, ) are the following: amples. ppling the istance Formula How far is the subwa ride from ak to mphon? ound to the nearest tenth. ach unit represents mile. ak has coordinates (, ). et (, ) represent ak. mphon has coordinates (, ). et (, ) represent mphon... W X X M Y Y ail otetaking Guide ame lass ate esson - esson bjectives Find the distance between two points in the coordinate plane Find the coordinates of the midpoint of a segment in the coordinate plane d d ( ) 00 trand: Measurement opic: Measuring hsical ttributes d %( ( )) ( ( )) ubstitute. d % ( ) ( ) M, "( ) ( ) he oordinate lane Use the istance Formula. implif within parentheses. orth Jackson entral (, ) edar mphon it laza ak (0, 0) (, ) outh lm. a. onstruct F with m F m. ) b. plain how ou can use our protractor to check that Y is the angle bisector of XYZ. Measure XY and YZ to see that the are congruent. X Y. raw. onstruct its perpendicular bisector. ). If m JK 0, then find m K and m JK. K bisects JK. J K m K 0, m JK 00 Z ail otetaking Guide Geometr esson - ame lass ate Finding the Midpoint has endpoints (, ) and (, ). Find the coordinates of its midpoint M. Use the Midpoint Formula. et (, ) be (, ) and (, ) be (, ). he midpoint has coordinates, ( ). ( ) he -coordinate is ( ) he -coordinate is he coordinates of the midpoint M are (, ). F Midpoint Formula Finding an ndpoint he midpoint of G is M(, ). ne endpoint is (, ). Find the coordinates of the other endpoint G. Use the Midpoint Formula. et (, ) be (, ) and the midpoint a, b be (, ). olve for and, the coordinates of G. Find the -coordinate of G. d Use the Midpoint Formula. d Multipl each side b. d implif. he coordinates of G are (, ). ubstitute for and for. implif. ubstitute for and for. implif. 0 d % www % 0 implif. 0. Use a calculator. o the nearest tenth, the subwa ride from ak to mphon is. miles. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

110 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Quick heck.. a. Using the map in ample, find the distance between lm and mphon. about. miles esson - esson bjectives Find perimeters of rectangles and squares, and circumferences of circles Find areas of rectangles, squares, and circles erimeter, ircumference, and rea 00 trand: Measurement opic: Measuring hsical ttributes Ke oncepts. 0 b. Maple is located miles west and miles north of it laza. Find the distance between edar and Maple. about. miles. Find the coordinates of the midpoint of XY with endpoints X(, ) and Y(, ). (, ). he midpoint of XY has coordinates (, ). X has coordinates (, ). Find the coordinates of Y. (, ) Geometr esson - ail otetaking Guide ame lass ate Finding erimeter in the oordinate lane Quadrilateral has vertices (0, 0), (, ), (, ), and (, 0). Find the perimeter. raw and label on a coordinate plane. Find the length of each side. dd the lengths to find the perimeter. %( 0 ) ( 0 ) www % Use the istance Formula. % wwwww % u u u u uler ostulate %( ) ( 0 ) Use the istance Formula. %( ) % wwwww % 0 erimeter erimeter erimeter he perimeter of quadrilateral is units. u u u u uler ostulate Finding rea of a ircle Find the area of in terms of π. In, r. d. π r Formula for the area of a circle.. π ( ) ubstitute for r d erimeter and rea s s quare with side length s. erimeter s rea s b h h b ectangle with base b and height h. erimeter b h rea bh ostulate - If two figures are congruent, then their areas are equal. ostulate -0 he area of a region is the sum of the areas of its non-overlapping parts. amples. Finding ircumference G has a radius of. cm. Find the circumference of G in terms of π. hen find the circumference to the nearest tenth. p r Formula for circumference of a circle.. π( ) ubstitute for r. π act answer 0.00 Use a calculator. he circumference of G is p, or about 0. cm. d r ircle with radius r and diameter d. ircumference pd or pr rea pr. cm G ail otetaking Guide Geometr esson - ame lass ate Finding rea of an Irregular hape Find the area of the figure to the right. raw a horizontal line to separate the figure into three squares non-overlapping figures: one rectangle and two. Find each area. hen add the areas. bh d Use the formulas. s ( )( ) d ubstitute. ( ) d implif. dd the areas. implif. he area of the figure is ft. Quick heck.. a. Find the circumference of a circle with a radius of m in terms of π. p m b. Find the circumference of a circle with a diameter of m to the nearest tenth..m. Graph quadrilateral KM with vertices K(, ), (, ), M(, ), and (, ). Find the perimeter of KM. 0 units. You are designing a rectangular banner for the front of a museum. he banner will be ft wide and d high. How much material do ou need in square ards? d. op the figure in ample. eparate it in a different wa. Find the area. 0 ft ft (, ) K(, ) ft ft ft ft ft M(, ) (, ) 0 ft. π he area of is.p d. ft Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

111 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives ecognize conditional statements Write converses of conditional statements 00 trand: Geometr opic: Mathematical easoning onditional tatements Finding a ountereample Find a countereample to show that this conditional is false: If 0, then 0. countereample is a case in which the hpothesis is true and the conclusion is false. his countereample must be an eample in which 0 (hpothesis true) and 0 or 0 (conclusion false). Vocabular and Ke oncepts. ecause an negative number has a positive square, one possible countereample is. onditional tatements and onverses tatement ample mbolic Form You read it onditional If an angle is a straight angle, p q If p, then its measure is 0. then q. onverse If the measure of an angle is q p If q, 0, then it is a straight angle. then p. conditional is an if-then statement. he hpothesis is the part that follows if in an if-then statement. he conclusion is the part of an if-then statement (conditional) that follows then. he truth value of a statement is true or false according to whether the statement is true or false, respectivel. he converse of the conditional if p, then q is the conditional if q, then p. amples. Identifing the Hpothesis and the onclusion Identif the hpothesis and conclusion: If two lines are parallel, then the lines are coplanar. In a conditional statement, the clause after if is the hpothesis and the clause after then is the conclusion. Hpothesis: wo lines are parallel. onclusion: he lines are coplanar. Geometr esson - ail otetaking Guide ame lass ate ample. Finding the ruth Value of a onverse Write the converse of the conditional. hen determine the truth value of each. If a, then a. onditional: If a, then a. he converse echanges the hpothesis and the conclusion. onverse: If a, then a. he conditional is false. countereample is a : ( ), and. ecause, the converse is true. Quick heck.. Write the converse of the following conditional: If two lines are not parallel and do not intersect, then the are skew. If two lines are skew, then the are not parallel and do not intersect.. Write the converse of each conditional. etermine the truth value of the conditional and its converse. (Hint: ne of these conditionals is not true.) a. If two lines do not intersect, then the are parallel. onverse: If two lines are parallel, then the do not intersect. he conditional is false and the converse is true. b. If, then u u. onverse: If u u, then. ecause ( ), which is greater than 0, the hpothesis is true. ecause 0, the conclusion is false. he countereample shows that the conditional is false. Writing the onverse of a onditional Write the converse of the following conditional. If, then. he converse of a conditional echanges the hpothesis and the conclusion. onditional onverse Hpothesis onclusion Hpothesis onclusion o the converse is: If, then. Quick heck.. Identif the hpothesis and the conclusion of this conditional statement: If, then. Hpothesis: onclusion:. how that this conditional is false b finding a countereample: If the name of a state includes the word ew, then the state borders an ocean. ountereample: ew Meico does not border an ocean. It is a countereample, so the conditional is false. ail otetaking Guide Geometr esson - ame lass ate esson - esson bjectives Write biconditionals ecognize good definitions Vocabular and Ke oncepts. 00 trand: Geometr opics: imension and hape; Mathematical easoning iconditional tatements biconditional combines p q and q p as p q. tatement ample mbolic Form You read it iconditional n angle is a straight angle if p q p if and q and onl if its measure is 0. onl if. biconditional statement is the combination of a conditional statement and its converse. biconditional contains the words if and onl if. amples. Writing a iconditional onsider the true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. onditional: If, then 0. o write the converse, echange the hpothesis and conclusion. onverse: If 0, then. When ou subtract from each side to solve the equation, ou get. ecause both the conditional and its converse are true, ou can combine them in a biconditional using the phrase if and onl if. iconditional: if and onl if 0. eparating a iconditional into arts Write the two statements that form this biconditional. iconditional: ines are skew if and onl if the are noncoplanar. Write a biconditional as two conditionals that are converses of each other. onditional: If lines are skew, then the are noncoplanar. iconditionals and efinitions he conditional is true and the converse is false. onverse: If lines are noncoplanar, then the are skew. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

112 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Writing a efinition as a iconditional how that this definition of triangle is reversible. hen write it as a true biconditional. efinition: triangle is a polgon with eactl three sides. he original conditional is true. onditional: If a polgon is a triangle, then it has eactl three sides.. Write two statements that form this biconditional about integers greater than : number is prime if and onl if it has onl two distinct factors, and itself. tatement: If a number is prime, then it has onl two distinct factors, and itself. he converse is also true. onverse: If a polgon has eactl three sides, then it is a triangle. ecause both statements are true, the can be combined to form a biconditional. tatement: If a number has onl two distinct factors, and itself, then it is prime. iconditional: polgon is a triangle if and onl if it has eactl three sides. Identifing a Good efinition Is the following statement a good definition? plain. n apple is a fruit that contains seeds. he statement is true as a description of an apple. change n apple and a fruit that contains seeds. he converse reads: fruit that contains seeds is an apple. here are man fruits that contain seeds but are not apples, such as lemons and peaches. hese are countereamples, so the converse of the statement is false. he original statement is not a good definition because the statement is not reversible. Quick heck.. onsider the true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. onditional: If three points are collinear, then the lie on the same line. onverse: If three points lie on the same line, then the are collinear. he converse is true. iconditional: hree points are collinear if and onl if the lie on the same line. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Use the aw of etachment Use the aw of llogism Vocabular and Ke oncepts. 00 trand: Geometr opic: Mathematical easoning aw of etachment If a conditional is true and its hpothesis is true, then its In smbolic form: If p q is a true statement and p is true, then q is true. aw of llogism If p q and q r are true statements, then p r is a true statement. is true. eductive reasoning is a process of reasoning logicall from given facts to a conclusion. amples. Using the aw of etachment gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make? he first sentence contains a conditional statement. he hpothesis is it rains. ecause the hpothesis is true, the gardener can conclude that the garden will be watered. conclusion Using the aw of etachment For the given statements, what can ou conclude? Given: If is acute, then m 0. is acute. conditional and its hpothesis are both given as true. the aw of etachment, ou can conclude that the conclusion of the conditional, m 0, is true. eductive easoning. how that this definition of right angle is reversible. hen write it as a true biconditional. efinition: right angle is an angle whose measure is 0. onditional: If an angle is a right angle, then its measure is 0. onverse: If an angle has measure 0, then it is a right angle. he two statements are true. iconditional: n angle is a right angle if and onl if its measure is 0.. Is the following statement a good definition? plain. square is a figure with four right angles. It is not a good definition because a rectangle has four right angles and is not necessaril a square. ail otetaking Guide Geometr esson - ame lass ate Using the aw of llogism Use the aw of llogism to draw a conclusion from the following true statements: If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle. he conclusion of the first conditional is the hpothesis of the second conditional. his means that ou can appl the aw of llogism. he aw of llogism: If p q and q r are true statements, then p r is a true statement. o ou can conclude: If a quadrilateral is a square, then it is a rectangle. rawing onclusions Use the aws of etachment and llogism to draw a possible conclusion. If the circus is in town, then there are tents at the fairground. If there are tents at the fairground, then aul is working as a night watchman. he circus is in town. ecause the conclusion of the first statement is the hpothesis of the second statement, ou can appl the aw of llogism to write a new conditional: If the circus is in town, then aul is working as a night watchman. he third statement means that the hpothesis of the new conditional is true. You can use the aw of etachment to form the conclusion: aul is working as a night watchman. Quick heck.. uppose that a mechanic begins work on a car and finds that the car will not start. an the mechanic conclude that the car has a dead batter? plain. o, there could be other things wrong with the car, such as a fault starter.. If a baseball plaer is a pitcher, then that plaer should not pitch a complete game two das in a row. Vladimir uñez is a pitcher. n Monda, he pitches a complete game. What can ou conclude? nswers ma var. ample: Vladimir uñez should not pitch a complete game on uesda. 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 Geometr ll-in-ne nswers Version

113 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate. If possible, state a conclusion using the aw of llogism. If it is not possible to use this law, eplain wh. a. If a number ends in 0, then it is divisible b 0. If a number is divisible b 0, then it is divisible b. If a number ends in 0, then it is divisible b. esson - esson bjective onnect reasoning in algebra and geometr easoning in lgebra 00 trand: lgebra and Geometr opics: lgebraic epresentations; Mathematical easoning Ke oncepts. b. If a number ends in, then it is divisible b. If a number ends in, then it is divisible b. ot possible; the conclusion of one statement is not the hpothesis of the other statement.. Use the aw of etachment and the aw of llogism to draw conclusions. he Volga iver is in urope. If a river is less than 00 miles long, it is not one of the world s ten longest rivers. If a river is in urope, then it is less than 00 miles long. onclusion: he Volga iver is less than 00 miles long. onclusion: he Volga iver is not one of the world s ten longest rivers. Geometr esson - ail otetaking Guide ame lass ate amples. Justifing teps in olving an quation Justif each step used to solve for. Given: ddition ropert of qualit ubtraction ropert of qualit ivision ropert of qualit Justifing teps in olving an quation uppose that points,, and are collinear with point between points and. olve for if,, and. Justif each step. egment ddition ostulate ( ) ( ) ubstitution ropert of qualit implif. ubtraction ropert of qualit Quick heck.. Fill in each ) missing reason. Given: M bisects K. ) M bisects K Given ngle m M m KM efinition of isector ubstitution ropert of qualit ubtraction ropert of qualit ivision ropert of qualit M ( 0) K. ecall that the length of line is. Find the lengths of and b substituting into the epressions in the diagram. heck that. roperties of qualit ddition ropert If a b, then a c b c. ubtraction ropert If a b, then a c b c. Multiplication ropert If a b, then a c b c. a b ivision ropert If a b and c 0, then. c c efleive ropert a a. mmetric ropert If a b, then b a. ransitive ropert If a b and b c, then a c. ubstitution ropert If a b, then b can replace a in an epression. istributive ropert a(b c) ab ac. roperties of ongruence efleive ropert mmetric ropert If, then. If, then. ransitive ropert If and F, then F. If and, then. ail otetaking Guide Geometr esson - ame lass ate ample. Using roperties of qualit and ongruence ame the propert that justifies each statement. a. If and, then. he conclusion of the conditional statement is the same as the equation (given) after has been substituted for (given). he propert used is the ubstitution ropert of qualit. b. If, then. he conclusion of the conditional statement shows the result after is subtracted from each side of the equation in the hpothesis. he propert used is the ubtraction ropert of qualit. c. If Q, Q, and, then. Use the ransitive ropert of ongruence the first two parts of the hpothesis: If Q and Q, then. Use the ransitive ropert of ongruence and the third part of the hpothesis: If and, then. he propert used is the ransitive ropert of ongruence. Quick heck.. ame the propert of equalit or congruence illustrated. a. XY XY efleive ropert of ongruence b. If m and m, then m m ransitive or ubstitution ropert of qualit for for Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

114 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - roving ngles ongruent complementar angles supplementar angles esson bjectives rove and appl theorems about angles 00 trand: Geometr opic: elationships mong Geometric Figures Vocabular and Ke oncepts. and are complementar angles. wo angles are complementar angles if and are supplementar angles. wo angles are supplementar angles if heorem -: Vertical ngles heorem congruent Vertical angles are. and heorem -: ongruent upplements heorem If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. heorem -: ongruent omplements heorem If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. heorem - ll right angles are congruent. heorem - If two angles are congruent and supplementar, then each is a right angle. vertical angles and are vertical angles, as are and. Vertical angles are two angles whose sides form two pairs of opposite ras. Geometr esson - adjacent angles and are adjacent angles, as are and. djacent angles are two coplanar angles that have a common side and a common verte but no common interior points. ail otetaking Guide ame lass ate Quick heck.. efer to the diagram for ample. a. Find the measures of the labeled pair of vertical angles. 0 b. Find the measures of the other pair of vertical angles. c. heck to see that adjacent angles are supplementar ecall the proof of heorem -. oes the size of the angles in the diagram affect the proof? Would the proof change if and were acute rather than obtuse? plain. o, the size of the angles does not affect the proof or the truth of the theorem. the sum of their measures is 0. the sum of their measures is 0. theorem is a conjecture that is proven. paragraph proof is a convincing argument that uses deductive reasoning in which statements and reasons are connected in sentences. amples. Using the Vertical ngles heorem Find the value of. he angles with labeled measures are vertical angles. ppl the Vertical ngles heorem to find. 0 Vertical ngles heorem 0 ddition ropert of qualit 0 ubtraction ropert of qualit ivision ropert of qualit roving heorem - Write a paragraph proof of heorem - using the diagram at the right. tart with the given: and are supplementar, and are supplementar. the definition of supplementar angles, m m 0 and m m 0. substitution, m m m m. Using the ubtraction ropert of qualit, subtract m from each side. You get m m, or. ( ) ( 0) ail otetaking Guide Geometr esson - ame lass ate esson - esson bjectives Identif angles formed b two lines and a transversal rove and use properties of parallel lines Vocabular and Ke oncepts. 00 trand: Measurement opic: Measuring hsical ttributes ostulate -: orresponding ngles ostulate If a transversal intersects two parallel lines, then corresponding angles are congruent. heorem -: lternate Interior ngles heorem If a transversal intersects two parallel lines, then alternate interior angles are congruent. heorem -: ame-ide Interior ngles heorem If a transversal intersects two parallel lines, then same-side interior angles are supplementar. m m 0 heorem -: lternate terior ngles heorem If a transversal intersects two parallel lines, then alternate eterior angles are congruent. heorem -: ame-ide terior ngles heorem If a transversal intersects two parallel lines, then same-side eterior angles are supplementar. m m 0 t a t b / m roperties of arallel ines Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

115 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate transversal is a line that intersects two coplanar lines at two t / distinct points. m lternate interior angles are nonadjacent interior angles that lie on opposite sides of the transversal. and are ame-side interior angles are interior angles that lie on the same side alternate interior angles. of the transversal. and are same-side interior angles. orresponding angles are angles that lie on the same side of the transversal and in corresponding positions relative to the coplanar and lines. corresponding angles. two-column proof is a displa that shows column shows 0 the steps Geometr esson - and the second column the steps to prove a theorem. amples. ppling roperties of arallel ines In the diagram of afaette egional irport, the black segments are runwas and the gra areas are taiwas and terminal buildings. ompare and the angle vertical to. lassif the angles as alternate interior angles, same-side interior angles, or corresponding angles. he angle vertical to is between the runwa segments. is between the runwa segments and on the opposite side of the transversal runwa. ecause alternate interior angles are not adjacent and lie between the lines on opposite sides of the transversal, and the angle vertical to are alternate interior angles. he first shows the reason for each step. ail otetaking Guide ame lass ate esson - esson bjectives Use a transversal in proving lines parallel Vocabular and Ke oncepts. 00 trand: Measurement opic: Measuring hsical ttributes ostulate -: onverse of the orresponding ngles ostulate If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel. heorem -: onverse of the lternate Interior ngles heorem If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. heorem -: onverse of the ame-ide Interior ngles heorem If two lines and a transversal form same-side interior angles that are supplementar, then the two lines are parallel. heorem -: onverse of the lternate terior ngles heorem If two lines and a transversal form alternate eterior angles that are congruent, then the two lines are parallel. heorem -: onverse of the ame-ide terior ngles heorem If two lines and a transversal form same-side eterior angles that are supplementar, then the two lines are parallel. roving ines arallel / m m. / m If, then m. If and are supplementar, then m. If, then m. If and are supplementar, then m. Finding Measures of ngles In the diagram at right, m and p q. Find m and m. and the angle are corresponding angles. ecause m, m b the orresponding ngles ostulate. ecause and are adjacent angles that form a 0 ubtract from each side to find m. straight angle, m m b the ngle ddition ostulate. If ou substitute for m, the equation becomes m 0. Using lgebra to Find ngle Measures In the diagram, m. Find the values of a, b, and c. a lternate Interior ngles heorem c 0 lternate Interior ngles heorem a b c 0 ngle ddition ostulate 0 0 b ubstitution ropert of qualit b ubtraction ropert of qualit Quick heck.. Use the diagram in ample. lassif and as alternate interior angles, same-side interior angles, or corresponding angles. same-side interior angles. Using the diagram in ample find the measure of each angle. Justif each answer. a. ; Vertical angles are congruent b. ; orresponding angles are congruent c. ; ame-side interior angles are supplementar d. ; orresponding angles are congruent e. ; lternate interior angles are congruent f. ; Vertical angles are congruent. Find the values of and. hen find the measures of the four angles in the trapezoid., ; 0, 0,, a b c ail otetaking Guide Geometr esson - 0 p q / m ( 0) ame lass ate flow proof uses arrows to show the logical connections between the statements. easons are written below the statements. amples. Using ostulate - Use the diagram at the right.which lines, if an, must be parallel if and are supplementar? Justif our answer with a theorem or postulate. It is given that and are supplementar. he diagram shows that and are supplementar. ecause supplements of the same angle are congruent (ongruent upplements heorem), ). ecause and are congruent corresponding angles, ) K b the onverse of the orresponding ngles ostulate. Using lgebra Find the value of for which m. he labeled angles are alternate interior angles. If m, the alternate interior angles are congruent, and their measures are equal. Write and solve the equation. 0 dd to each side. 0 ubtract from each side. 0 ivide each side b. ( ) K / ( ) m / m Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

116 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Quick heck.. Use the diagram from ample. Which lines, if an, must be parallel if? plain. u K u ; onverse of orresponding ngles ostulate esson - esson bjectives elate parallel and perpendicular lines arallel and erpendicular ines 00 trand: Geometr opic: elationships mong Geometric Figures. Find the value of for which a b. plain how ou can check our answer. ; (), and 0 Geometr esson - a b ( ) ail otetaking Guide ame lass ate he sides are constructed so that the inner and outer edges of each side are parallel. nd if an inner edge on one side is parallel to the outer edge on the same side, and at the same time the outer edge is parallel to the outer edge on the opposite side, then b heorem - each inner edge is parallel to the outer edge on the opposite side. ut that outer edge is parallel to the inner edge on the same side, so again b heorem -, inner edges on opposite sides are parallel. Using heorem - Write a paragraph proof. k Given: In a plane, k l and k m. lso, m 0. m rove: he transversal is perpendicular to line m. ince m 0, the transversal is perpendicular to line k. ince k m, b heorem - the transversal is also perpendicular to line m. Quick heck.. an ou assemble the framing at the right into a frame with opposite sides parallel? plain. Yes; From what is given in ample, can ou also conclude that the transversal is perpendicular to line l? Yes, b heorem - and the fact that k l Ke oncepts. heorem - If two lines are the are parallel parallel to the same line to each other., then heorem -0 In a plane, if two lines are perpendicular to the same line, m n then the are parallel to each other. heorem - In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. n m amples. eal-world onnection picture frame is assembled as shown. Given the book s eplanation for wh the outer edges on opposite sides of the frame are parallel, wh must the inner edges on opposite sides be parallel, too? a b c corners cut to form angles ail otetaking Guide Geometr esson - n a b. t m n. ame lass ate esson - esson bjectives lassif triangles and find the measures of their angles Use eterior angles of triangles Vocabular and Ke oncepts. heorem -: riangle ngle-um heorem he sum of the measures of the angles of a triangle is 0. m m m 0 heorem -: riangle terior ngle heorem he measure of each eterior angle of a triangle equals the sum of the measures of its two remote interior angles. n acute right n obtuse n equiangular triangle has triangle has triangle has triangle has three acute angles. one right angle. one obtuse angle. three congruent angles. n equilateral n isosceles scalene triangle has triangle has triangle has three congruent sides. at least two congruent sides. no congruent sides. n eterior angle of a polgon is an angle formed b a side and an etension of an adjacent side. 00 trand: Geometr opic: elationships mong Geometric Figures m terior angle arallel ines and the riangle ngle-um heorem emote interior angles are the two nonadjacent interior angles corresponding to each eterior angle of a triangle. emote interior angles Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

117 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. Quick heck. ppling the riangle ngle-um heorem In triangle, is a right angle, and. Find the values of a and c. Find c first, using the fact that is a right angle. m 0 efinition of a right angle c 0 a b. ritical hinking escribe how ou could use either or to find the value of b in ample. For, (0 0) 0 b 0. hen 0 b 0 and b 0. For, 0 0 b 0. hen 0 b 0 and b 0. c 0 0 ngle ddition ostulate 0 c 0 ubtract from each side. o find a, use. a m c 0 riangle ngle-um heorem m 0 efinition of perpendicular lines a ubstitute for m and for c. 0 0 a implif. 0 0 a ubtract from each side. ppling the riangle terior ngle heorem plain what happens to the angle formed b the back of the chair and the armrest as ou make a lounge chair recline more. ack rm he eterior angle and the angle formed b the back of the chair and the armrest are adjacent angles, which together form a straight angle. s one measure increases, the other measure decreases. he angle formed b the back of the chair and the armrest increases as ou make a lounge chair recline more. Geometr esson - (0 0 ) (0 0 ) ail otetaking Guide ame lass ate esson - esson bjectives lassif polgons Find the sums of the measures of the interior and eterior angles of polgons Vocabular and Ke oncepts. heorem -: olgon ngle-um heorem he sum of the measures of the angles of an n-gon is (n )0. heorem -: olgon terior ngle-um heorem he sum of the measures of the eterior angles of a polgon, one at each verte, is 0. For the pentagon, m m m m m 0. polgon is a closed plane figure with at least three sides that are segments. he sides intersect onl at their endpoints, and no two adjacent sides are collinear. polgon conve polgon does not have diagonal points outside of the polgon. concave polgon has at least one diagonal with points outside of the polgon. ot a polgon not a closed 00 trand: Geometr opic: elationships mong Geometric Figures ; figure he olgon ngle-um heorems ot a polgon two sides intersect between endpoints. Y iagonal M W K G Q ; conve polgon concave polgon. a. Find m. 0 b. ritical hinking Is it true that if two acute angles of a triangle are complementar, then the triangle must be a right triangle? plain. rue, because the measures of the two complementar angles add to 0, leaving 0 for the third angle. ail otetaking Guide Geometr esson - ame lass ate n equilateral polgon has all sides congruent. n equiangular polgon has all angles congruent. regular polgon is both equilateral and equiangular. amples. lassifing olgons lassif the polgon at the right b its sides. Identif it as conve or concave. tarting with an side, count the number of sides clockwise around the figure. ecause the polgon has sides, it is a dodecagon. hink of the polgon as a star. raw a diagonal connecting two adjacent points of the star. hat diagonal lies outside the polgon, so the dodecagon is concave. Finding a olgon ngle um Find the sum of the measures of the angles of a decagon. decagon has 0 sides, so n 0. um (n )(0) olgon ngle-um heorem 0 0 ( )(0) ubstitute for n.? 0 ubtract. 0 implif. Using the olgon ngle-um heorem Find m X in quadrilateral XYZW. he figure has sides, so n. m X m Y m Z m W ( )(0) olgon ngle-um heorem m X m Y ubstitute. 0 0 m X m Y implif. 0 0 m X m Y ubtract from each side. m X 0 m X m X ubstitute for m Y. m X 0 implif. m X ivide each side b. Y X Z 00 W 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

118 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate ppling heorem - regular heagon is inscribed in a rectangle. plain how ou know that all the angles labeled have equal measures. he heagon is regular, so all its angles are congruent. n eterior angle is the supplement of a polgon s angle because the are adjacent angles that form a straight angle. ecause supplements of congruent angles are congruent, all the angles marked have equal measures. Quick heck.. lassif each polgon b its sides. Identif each as conve or concave. a. b. heagon; conve. a. Find the sum of the measures of the angles of a -gon. 0 b. ritical hinking he sum of the measures of the angles of a given polgon is 0. How can ou use um (n )0 to find the number of sides in the polgon? You can solve the equation (n )0 0.. entagon has congruent angles. Find the measure of each angle. 0. a. In the figure from ample, find m b using the olgon terior ngle-um heorem. 0 b. Find m. Is an eterior angle? plain. 0; no, it is not formed b etending one side of the polgon. Geometr esson - octagon; concave ail otetaking Guide ame lass ate ransforming to lope-intercept Form ransform the equation to slope-intercept form. hen graph the resulting equation. tep ransform the equation to slope-intercept form. dd to each side. ivide each side b. implif. he -intercept is and the slope is. tep Use the -intercept and the slope to plot two points and draw the line containing them. Using oint-lope Form Write an equation in point-slope form of the line with slope that contains (, ). m ( ) Use point-slope form. (, ) ( ) implif. ( ) ( ) ubstitute for m and for (, ). (, ) (0, ) Quick heck.. Graph each equation. a. b. c. esson - esson bjectives Graph lines given their equations Write equations of lines ines in the oordinate lane 00 trand: lgebra opics: atterns, elations, and Functions; lgebraic epresentations Vocabular. lope-intercept form he slope-intercept form of a linear equation is m b. he standard form of a linear equation is. he point-slope form for a nonvertical line is tandard form m( ). ( ) oint-slope form amples. Graphing ines Using Intercepts Use the -intercept and -intercept to graph 0. o find the -intercept, substitute o find the -intercept, substitute 0 for and solve for. 0 for and solve for. 0 0 ( 0 ) 0 ( 0 ) he -intercept is. he -intercept is. point on the line is (,0). point on the line is (0, ). lot (, 0) and (0, ). raw the line containing the two points. (, 0) (0, ) Geometr esson - ail otetaking Guide ame lass ate ample. Writing an quation of a ine Given wo oints Write an equation in point-slope form of the line that contains the points G(, ) and H(, ). tep Find the slope. m Use the formula for slope. ( ) m ubstitute (, ) for (, ) and (, ) for (, ). 0 implif. tep elect one of the points. Write the equation in point-slope form. m ( ) oint-slope form. (, ) ( ) ( ) ubstitute for m and for (, ). ( ) implif. Quick heck.. Write an equation of the line with slope that contains the point (, ). ( ). Write an equation of the line that contains the points (, 0) and Q(, ). 0 ( ) or ( ) Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

119 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives elate slope and parallel lines elate slope and perpendicular lines lopes of arallel and erpendicular ines 00 trand: Measurement opic: Measuring hsical ttributes Writing quations of arallel ines Write an equation in point-slope form for the line parallel to that contains (, ). tep o find the slope of the line, rewrite the equation in slope-intercept form. ubtract from each side. Ke oncepts. ivide each side b. lopes of arallel ines If two nonvertical lines are parallel, their slopes are equal. If the slopes of two distinct nonvertical lines are equal, the lines are parallel. n two vertical lines are parallel. lopes of erpendicular ines If two nonvertical lines are perpendicular, the product of their slopes is. If the slopes of two lines have a product of, the lines are perpendicular. n horizontal line and vertical line are perpendicular. amples. etermining Whether ines are arallel re the lines and parallel? plain. he equation is in slope-intercept form. Write the equation in slope-intercept form. ubtract from each side. ivide each side b. Geometr esson - he line has slope. he line has slope. he lines are not parallel because their slopes are not equal. ample. Writing quations for erpendicular ines Write an equation for a line perpendicular to that contains (0, 0). tep o find the slope of the given line, rewrite the equation in slope-intercept form. ubtract from each side. ivide each side b. he line has slope. ail otetaking Guide ame lass ate tep Find the slope of a line perpendicular to. et m be the slope of the perpendicular line. m he product of the slopes of perpendicular lines is. m? Multipl each side b. ( ) m implif. tep Use point-slope form, m( ), to write an equation for the new line. 0 0 (0, 0) ( ) ubstitute for m and for (, ). 0 ( ) implif. Quick heck.. Write an equation for the line perpendicular to 0 that contains (, ). ( ) he line has slope. tep Use point-slope form to write an equation for the new line. m ( ) (, ) ( ) implif. ( ) ( ( )) ubstitute for m and for (, ). Quick heck.. re the lines parallel? plain. a. and b. and 0 Yes; ach line has slope o; the lines have the same slope and and the -intercept, so the are the same line. -intercepts are different.. Write an equation for the line parallel to that contains (, ). ( ) ail otetaking Guide Geometr esson - ame lass ate esson - esson bjectives onstruct parallel lines onstruct perpendicular lines ample. onstructing m amine the diagram at right. plain how to construct congruent to H. onstruct the angle. Use the method learned for constructing congruent angles. tep With the compass point on point H, draw an arc that intersects the sides of H. tep With the same compass setting, put the compass point on point. raw an arc. tep ut the compass point below point where the arc * ) intersects H. pen the compass to the length where the arc intersects line. Keeping the same compass setting, put the compass point above point where the * ) arc intersects H. raw an arc to locate a point. tep Use a straightedge to draw line m through the point ou located and point. Quick heck.. Use ample. plain wh lines and m must be parallel. If corresponding angles are congruent, the lines are parallel b the onverse of orresponding ngles ostulate. onstructing arallel and erpendicular ines 00 trand: Geometr opic: elationships mong Geometric Figures H J m / Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

120 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate ample. onstructing a pecial Quadrilateral onstruct a quadrilateral with both pairs of sides parallel. tep raw point and two ras with endpoints at. abel point on one ra and point on the other ) ra. tep onstruct a ra parallel to through point. 0 ) tep onstruct a ra parallel to through point. tep abel point ) ) where the ra parallel to intersects the ra parallel to. Quadrilateral has both pairs of opposite sides parallel. Quick heck.. raw two segments. abel their lengths c and d. onstruct a quadrilateral with one pair of parallel sides of lengths c and d. c d Geometr esson - esson - esson bjective ecognize congruent figures and their corresponding parts Vocabular and Ke oncepts. heorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. ongruent polgons are polgons that have corresponding sides congruent and corresponding angles congruent. amples. aming ongruent arts QJ. ist the congruent corresponding parts. ist the corresponding sides and angles in the same order. ngles: Q J ides: > Q J QJ c d d F ail otetaking Guide ame lass ate 00 trand: Geometr opic: ransformation of hapes and reservation of roperties Using ongruenc XYZ KM, m Y, and m M. Find m X. Use the riangle ngle-um heorem and the definition of congruent polgons to find m X. m X m Y m Z 0 riangle ngle-um heorem m Z m M orresponding angles of congruent triangles are congruent. m Z ubstitute for m M. F ongruent Figures I F Q J H GHI G F ample. erpendicular From a oint to a ine amine the * ) construction. t what special point does G meet line? / oint is the same distance from point as it is from point F because the arc was made with one compass opening. oint G is the same distance from point as it is from point F G because both arcs were made with the same compass opening. * ) his means that G intersects line at the midpoint of F, and * ) that G is the perpendicular bisector of F. Quick heck. * ) * ) * ) * ). a. Use a straightedge to draw F. onstruct FG so that FG F at point F. * X ) G F * X ) b. raw a line and a point Z not on. onstruct so that. Z X ail otetaking Guide Geometr esson - ame lass ate Finding ongruent riangles an ou conclude that? ist corresponding vertices in the same order. If, then. he diagram above shows, not. he statement is not true. otice that,, and. lso, and. Using heorem -, ou can conclude that. ince all of the corresponding sides and angles are congruent, the triangles are congruent. he correct wa to state this is. Quick heck.. WY MKV. ist the congruent corresponding parts. Use three letters for each angle. ides: ngles: WY MK lwy lmvk W MV lwy lvmk Y KV lwy lmkv * Z ) * Z ). It is given that WY MKV. If m Y, what is m K? plain. m K. an ou conclude that JK M? Justif our answer. J K orresponding angles of congruent triangles are congruent and have the same measure. o. he corresponding sides are not necessaril equal. M * X ) F m X 0 ubstitute. m X 0 implif. m X ubtract from each side. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

121 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate ample. roving riangles ongruent Given: G G,,, G # rove: G G G esson - esson bjective rove two triangles congruent using the and ostulates riangle ongruence b and 00 trand: Geometr opic: ransformation of hapes and reservation of roperties tatements easons Ke oncepts.. G > G. Given. >. Given. G > G. efleive ropert of ongruence.. Given. G G. ight angles are congruent. G G. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent (heorem - ).. G G. efinition of congruent triangles Quick heck.. Given:,,,, rove:.... ; Geometr esson - tatements ; ;.... Given Given easons Vertical angles are congruent. efinition of ongruent riangles ail otetaking Guide ame lass ate Using >. What other information do ou need to prove b? It is given that >. lso, > b the efleive ropert of ongruence. You now have two pairs of corresponding congruent sides.herefore, if ou know, ou can prove b. re the riangles ongruent? From the information given, can ou prove G H? plain. Given: G H, G > H It is given that G H and G > H. G H b the efleive ropert of ongruence. wo pairs of corresponding sides and their included angles are congruent, so G H b the ostulate. F J Quick heck.. Given: HF HJ, FG JK, H is the midpoint of GK. rove: FGH JKH G H K tatements easons. HF > HJ, FG > JK. Given. H is the midpoint of GK.. Given. GH > HK. efinition of Midpoint. FGH JKH. ostulate. What other information do ou need to prove b?. From the information given, can ou prove? plain. Given: >, > o; ou don t know that or that. ostulate -: ide-ide-ide () ostulate If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are G congruent. ostulate -: ide-ngle-ide () ostulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. GHF Q F amples. roving riangles ongruent Given: M is the midpoint of XY, X > Y rove: MX MY Write a paragraph proof. You are given that M is the midpoint of XY, and X Y. Midpoint M implies that MX MY. M > M b the efleive ropert of ongruence, so MX MY b the ostulate. ail otetaking Guide Geometr esson - H F ame lass ate esson - esson bjective rove two triangles congruent using the ostulate and the heorem Ke oncepts. ostulate -: ngle-ide-ngle () ostulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. HG K G ample. Using uppose that F is congruent to and I is not congruent to. ame the triangles that are congruent b the ostulate. he diagram shows and F > > G. If F, then F G. herefore, FI G b. 00 trand: Geometr opic: ransformation of hapes and reservation of roperties heorem -: ngle-ngle-ide () heorem If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. M XG G Quick heck.. Using onl the information in the diagram, can ou conclude that IF is congruent to either of the other two triangles? plain. X G Q F M K H M X Y riangle ongruence b and F I o; nl one angle and one side are shown to be congruent. t least one more congruent side or angle is necessar to prove congruence with,, or. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

122 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. Quick heck. Writing a roof Using Write a paragraph proof.. Write a proof. Given: / > /, > rove: X Y It is given that and >. X Y b the Vertical ngles heorem. ecause two pairs of corresponding angles and their included sides are congruent, X Y b. X Y Given: M >, M rove: M It is given that M and M. M because vertical angles are congruent. M b. M lanning a roof Write a plan for a proof that uses. Given:, rove: ecause, b the lternate Interior ngles heorem. hen if a pair of corresponding sides are congruent. the efleive ropert,, so b. Writing a roof Write a two-column proof that uses. Given:, rove: tatements easons.,. Given. l l. If lines are parallel, then. angles are congruent. alternate interior. >. efleive ropert of ongruence.. heorem Geometr esson - ail otetaking Guide ame lass ate esson - esson bjective Use triangle congruence and to prove that parts of two triangles are congruent Vocabular. stands for: orresponding arts of ongruent riangles are ongruent. amples. ongruence tatements he diagram shows the frame of an umbrella. What congruence statements besides can ou prove from the diagram, in which > and are given? b the efleive ropert of ongruence, and b. because corresponding parts of congruent triangles are congruent. When two triangles are congruent, ou can form congruence statements about three pairs of corresponding angles and three pairs of corresponding sides. ist the congruence statements. ides: > Given efleive ropert of ongruence ngles: l l ther congruence statement Given orresponding arts of ongruent riangles ther congruence statement he congruence statements that remain to be proved are l l and. Using ongruent riangles: 00 trand: Geometr opic: ransformation of hapes and reservation of roperties ib haft tretcher. Write a flow proof. Given: Q, bisects Q. rove: Q Q Q Given bisects Q Given Q efinition of angle bisector k kq. ecall ample. plain how ou could prove using. l l. It is given that You know that l l b the l l lternate Interior ngles heorem. heorem -,. the efleive ropert of ongruence,. herefore, k k b. efleive ropert of ongruence ail otetaking Guide Geometr esson - ame lass ate Using ight riangles ccording to legend, one of apoleon s followers used congruent triangles to estimate the width of a river. n the riverbank, the officer stood up straight and lowered the visor of his cap until the farthest thing he could see was the edge of the opposite bank. He then turned and noted the spot on his side of the river that was in line with his ee and the tip of his visor. F Given: G and F are right angles; G F. he officer then paced off the distance to this spot and declared that distance to be the width of the river! he given states that G and F are right angles. What conditions must hold for that to be true? G and F are the angles that the officer makes with the ground. o the officer must stand perpendicular to the ground, and the ground must be level or flat. Quick heck.. Given: Q, Q rove: Q > G It is given that Q, Q. b the efleive ropert of. kq k b. Q b.. ecall ample. bout how wide was the river if the officer paced off 0 paces and each pace was about feet long? 0 feet Q 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 Geometr ll-in-ne nswers Version

123 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective Use and appl properties of isosceles triangles Vocabular and Ke oncepts. heorem -: Isosceles riangle heorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. heorem -: onverse of Isosceles riangle heorem If two angles of a triangle are congruent, then the sides opposite those sides are congruent. heorem - he bisector of the verte angle of an isosceles triangle is the perpendicular bisector of the base. ' and bisects. orollar to heorem - If a triangle is equilateral, then the triangle is equiangular. X Y Z orollar to heorem - If a triangle is equiangular, then the triangle is equilateral. XY YZ ZX corollar is a statement that follows immediatel from a theorem. Geometr esson - Isosceles and quilateral riangles 00 trand: Geometr opic: elationships mong Geometric Figures X X Y Y Z Z ail otetaking Guide ame lass ate m Given m m m M m M 0 ransitive ropert of qualit riangle-um heorem 0 0 ubstitute. 0 0 implif. 0 0 ubtract from each side. ivide each side b. herefore, 0 and. Using Isosceles riangles he garden shown at the right is in the shape of a regular heagon. uppose that a segment is drawn between the endpoints of the angle marked. Find the angle measures of the triangle that is formed. he measure of is 0. ecause the garden is a regular heagon, the sides have equal length, so the triangle is isosceles. the Isosceles riangle heorem, the unknown angles are congruent. he measure of the angle marked is 0, and the sum z of the angle measures of a triangle is 0. abel each unknown angle : flagstone walk 0 0 o the angle measures of the triangle are 0, 0, and 0. Quick heck.. uppose m. Find the values of and. 0,. Use the diagram from ample. he path around the garden is made up of rectangles and equilateral triangles. What is the measure of z, the angle at the outside corner of the path? 0 M raised bed garden railroad tie steps he legs of an isosceles triangle are the congruent sides. he base of an isosceles triangle is the third, or non-congruent, side. ample. Using the Isosceles riangle heorems plain wh is isosceles. and X are alternate interior angles formed b X, *, and the transversal. ecause X ), X X. he diagram shows that X. the ransitive ropert of ongruence,. You can use the onverse of the Isosceles riangle heorem to conclude that >. the definition of an isosceles triangle, is isosceles. Quick heck.. an ou deduce that UV is isosceles? plain. o; neither luv nor lvu can be shown to be congruent l. to amples. Using lgebra uppose that m. Find the values of and. M ' he bisector of the verte angle of an isosceles triangle is the perpendicular bisector of the base. 0 m m Verte ngle ase ngles he verte angle of an isosceles triangle is formed b the two congruent sides (legs). he base angles of an isosceles triangle are formed b the base and the legs. ail otetaking Guide Geometr esson - egs ase efinition of perpendicular Isosceles riangle heorem U V M W ame lass ate esson - esson bjective rove triangles congruent using the H heorem Vocabular and Ke oncepts. heorem -: Hpotenuse-eg (H) heorem If the hpotenuse and a leg of one right triangle are congruent to the hpotenuse and leg of another right triangle, then the triangles are congruent. he hpotenuse of a right triangle is its longest side, or the side opposite the right angle. he legs of a right triangle are the two shortest sides, or the sides that are not opposite the right angle. amples. roving riangles ongruent ne student wrote M b the H heorem for the diagram. Is the student correct? plain. he diagram shows the following congruent parts. > M lm ince is the hpotenuse and is a leg of right triangle, and M is the hpotenuse and is a leg of right triangle M, the triangles are congruent b the H heorem. he student is correct. egs 00 trand: Geometr opic: ransformation of hapes and reservation of roperties Hpotenuse ongruence in ight riangles M Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

124 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Flow roof Using the H heorem XYZ is isosceles. From verte X, a perpendicular is drawn to YZ, intersecting YZ at point M. plain wh XMY XMZ. X Quick heck.. M Q wo-olumn roof Using the H heorem Given: and are right angles, > rove: tatements easons. and are. Given right angles. and. efinition of ight riangle are right triangles. >. Given.. efleive ropert of ongruence.. If the hpotenuse and a leg of one right triangle are congruent to the hpotenuse and leg of another right triangle, then the triangles are congruent (H heorem). Y M Z XM YZ XY Given XZ efinition of isosceles Geometr esson - XMY and XMZ are right angles efinition of perpendicular lines XM XM efleive ropert of ongruence XMY and XMZ are right triangles efinition of right triangle XMY XMZ H heorem ail otetaking Guide ame lass ate esson - esson bjectives Identif congruent overlapping triangles rove two triangles congruent b first proving two other triangles congruent 00 trand: Geometr opic: elationships mong Geometric Figures amples. Identifing ommon arts ame the parts of the sides that FG and HG share. Identif the overlapping triangles. arts of sides G and G are shared b FG and HG. hese parts are HG and FG, respectivel. Using ommon arts Write a plan for a proof that Z does not use overlapping triangles. M Given: ZXW YWX, ZWX YXW rove: ZW > YX W abel point M where ZX intersects WY. ZW > YX b if ZWM YXM. You can prove these triangles congruent using as follows: ook at MWX. MW > MX b the onverse of the Isosceles riangle heorem. ZMW YMX because vertical angles are congruent. the ngle ddition ostulate, ZWM YXM. o ZWM YXM b. o ZW > YX b. Quick heck.. he diagram shows triangles from the scaffolding that workers used when the repaired and cleaned the tatue of ibert. a. ame the common side in and. b. ame another pair of triangles that share a common side. ame the common side. Using orresponding arts of ongruent riangles H G F F Y X Which two triangles are congruent b the H heorem? Write a correct congruence statement. M Q. Write a paragraph proof of ample. It is given that XM is perpendicular to YZ. his means that lymx and ZMX are right angles, since that is the definition of perpendicular lines. It follows, then, that YMX and ZMX are right triangles, since the each contain a right angle. It is given that XYZ is isosceles, so XY XZ b the definition of isosceles. XM XM b the efleive ropert of ongruence. herefore, since the hpotenuses are congruent and one leg is congruent, XMY XMZ b the H heorem.. You know that two legs of one right triangle are congruent to two legs of another right triangle. plain how to prove the triangles are congruent. ince all right angles are congruent, the triangles are congruent b. ail otetaking Guide Geometr esson - ame lass ate amples. Using wo airs of riangles Write a paragraph proof. X Given: XW > YZ, XWZ and YZW are right angles. rove: XW YZ lan: XW YZ b if WXZ lzyw. hese angles are congruent b if XWZ YZW. hese triangles are congruent b. roof: You are given XW > YZ. ecause XWZ and YZW are W right angles, XWZ YZW. WZ > ZW b the efleive ropert of ongruence. herefore, XWZ YZW b. WXZ ZYW b, and XW YZ because vertical angles are congruent. herefore, XW YZ b. eparating verlapping riangles Given: >, > Write a two-column proof to show that. lan: b if. his congruence holds b if >. roof: tatement eason.. efleive ropert of ongruence.,. Given.,. ongruent sides have equal measure... ubtraction ropert of qualit.. egment ddition ostulate.. ubstitution. >. efinition of ongruence.... X Y Z nswers ma var. ample: and ; Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

125 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Quick heck.. Write a paragraph proof. Given: rove: > It is given that, so l l b. his means that is isosceles b the onverse of Isosceles riangle heorem. hen b the definition of isosceles triangle. esson - esson bjective Use properties of midsegments to solve problems Vocabular and Ke oncepts. Midsegments of riangles 00 trand: Geometr opics: elationships mong Geometric Figures heorem -: riangle Midsegment heorem Q. Write a two-column proof. Given: >, Q Q rove: Q Q tatement eason., lq lq. Given. Q Q. efleive ropert of ongruence. Q Q.. Q Q.. lq lq.. Q Q. efleive ropert of ongruence. Q Q.. Write a two-column proof. Given:, rove: > F 0 tatement l l, l l l lf.. F.. F. Geometr esson - F eason Given efleive ropert of ongruence Vertical ngles are ongruent ail otetaking Guide ame lass ate Identifing arallel egments Find m M and m M. M and are cut b transversal, so M and are corresponding angles. M b the riangle Midsegment heorem, so M b the orresponding ngles ostulate. m M because congruent angles have the same measure. In M, M, so m M m M b the Isosceles riangle heorem. m M b substitution. ppling the riangle Midsegment heorem ean plans to swim the length of the lake, as shown in the photo. plain how ean could use the riangle Midsegment heorem to measure the length of the lake. o find the length of the lake, ean starts at the middle of the left edge of the lake and paces straight along that end of the lake. He counts the number of strides (). Where the lake ends, he sets a stake. He paces the same number of strides () in the same direction and sets another stake. he first stake marks the midpoint of one side of a triangle. ean paces from the second stake straight to the middle of the other end of the lake. He counts the number of his strides (). ean finds the midsegment of the second side b pacing eactl half the number of strides back toward the second stake. He paces strides. From this midpoint of the second side of the triangle, he returns to the midpoint of the first side, counting the number of strides (). ean has paced a triangle. He has also formed a midsegment of a triangle whose third side is the length of the lake. the riangle Midsegment heorem, the segment connecting the two midpoints is half the distance across the lake. o, ean multiplies the length of the midsegment b to find the length of the lake.?? M If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length. midsegment of a triangle is a segment connecting the midpoints of two sides. coordinate proof is a form of proof in which coordinate geometr and algebra are used to prove a theorem. amples. Finding engths In XYZ, M,, and are midpoints. he perimeter of M is 0. Find and YZ. ecause the perimeter of M is 0, ou can find. M M 0 efinition of perimeter 0 ubstitute for M. for M and 0 implif. ubtract from each side. Use the riangle Midsegment heorem to find YZ. M YZ riangle Midsegment heorem YZ ubstitute for M. YZ Multipl each side b. ail otetaking Guide Geometr esson - ame lass ate Quick heck.. 0 and. Find,, and. ; 0; 0. ritical hinking Find m VUZ. Justif our answer. X U Y Z V ; UVXY so VUZ and YXZ are corresponding and congruent.. is a new bridge being built over a lake as shown. Find the length of the bridge. ft 0 ft ridge ft 0 ft M Y Z Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

126 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective Use properties of perpendicular bisectors and angle bisectors isectors in riangles 00 trand: Geometr opics: elationships mong Geometric Figures amples. ppling the erpendicular isector heorem Use the map of Washington,.. escribe the set of points that are equidistant from the incoln Memorial and the apitol. Vocabular and Ke oncepts. heorem -: erpendicular isector heorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. heorem -: onverse of the erpendicular isector heorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. heorem -: ngle isector heorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. heorem -: onverse of the ngle isector heorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the angle bisector. he distance from a point to a line is the length of the perpendicular segment from the point to the line. 0 Geometr esson - * ) * ) is in. from and. ail otetaking Guide ame lass ate Quick heck. * ). Use the information given in the diagram. is the perpendicular bisector of. Find and. plain our reasoning. * ) ; ; is the perpendicular bisector of ; therefore and. ) ). a. ccording to the diagram, how far is K from H? From? 0; 0 ) b. What can ou conclude about K? ) K is the angle bisector of lh. c. Find the value of. 0 d. Find m H. 0 K 0 ( 0) H he onverse of the erpendicular isector heorem states: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. point that is equidistant from the incoln Memorial and the apitol must be on the perpendicular bisector of the segment whose endpoints are the incoln Memorial and the apitol. Using the ngle isector heorem Find, F, and F in the diagram at right. ngle isector F F heorem ubstitute. F dd to each side. ubtract from each side.. ivide each side b... F ( ) ubstitute... F ( ) ubstitute. ail otetaking Guide Geometr esson - ame lass ate esson - esson bjective Identif properties of perpendicular bisectors and angle bisectors Identif properties of medians and altitudes of a triangle Vocabular and Ke oncepts. heorem - he perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. heorem - he bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. 00 trand: Geometr opics: elationships mong Geometric Figures heorem - he medians of a triangle are concurrent at a point that is two-thirds the distance from each verte to the midpoint of the opposite side. J G F FH H G heorem - J he lines that contain the altitudes of a triangle are concurrent. F oncurrent lines are three or more lines that meet in one point. he point of concurrenc is the point at which concurrent lines intersect. circle is circumscribed about a polgon when the vertices ircumscribed circle of the polgon are on the circle. he circumcenter of a triangle is the point of concurrenc of Q the perpendicular bisectors of a triangle. median of a triangle is a segment that Median has as its endpoints a verte of the Q oncurrent ines, Medians, and ltitudes triangle and the midpoint of the opposite side. circumcenter Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

127 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate circle is inscribed in a polgon if the sides of the polgon are tangent to the circle. he incenter of a triangle is the point of currenc of the angle bisectors of the triangle. n altitude of a triangle is the perpendicular segment from a verte to the line containing the opposite side. U Inscribed circle XI YI ZI Y X I Z V incenter ppling heorem - it planners want to locate a fountain equidistant from three straight roads that enclose a park. plain how the can find the location. he roads form a triangle around the park. heorem - states that the bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. Highwa 0 Mariposa oulevard ark ndover oad he cit planners should find the point of concurrenc of the he centroid of a triangle is the point of intersection of the medians of the triangle. he orthocenter of a triangle is the point of intersection of the lines containing the altitudes of the triangle. amples. Finding the ircumcenter Find the center of the circle that circumscribes XYZ. ecause X has coordinates (, ) and Y has coordinates (, ), XY lies on the vertical line. he perpendicular bisector of XY is the horizontal line that passes through (, ) or (, ), so the equation of the perpendicular bisector of XY is. ecause X has coordinates (, ) and Z has coordinates (, ), XZ lies on the horizontal line. he perpendicular bisector of XZ is the vertical line that passes through (, ) or (, ), so the equation of the perpendicular bisector of XZ is. raw the lines and. he intersect at the point (, ). his point is the center of the circle that circumscribes XYZ. Geometr esson - cute riangle ight riangle btuse riangle ltitude is ltitude is ltitude is inside. a side. outside. Y X (, ) Z ail otetaking Guide ame lass ate Quick heck.. a. Find the center of the circle that ou can circumscribe about the triangle with vertices (0, 0), (, 0), and (0, ). (, ) b. ritical hinking In ample, eplain wh it is not necessar to find the third perpendicular bisector. heorem -: ll the perpendicular bisectors of the sides of a triangle are concurrent.. a. he towns of damsville, rooksville, and artersville want to build a librar that is equidistant from the three towns. how on the diagram where the should build the librar. plain. raw segments connecting the towns. uild the librar at the intersection point of the perpendicular bisectors of the segments. b. What theorem did ou use to find the location? he perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices (heorem -).. Using the diagram in ample, find MX. heck that WM MX WX.. Is MY a median, an altitude, or neither? plain. median; MY is a segment drawn from verte M to the midpoint of the opposite side. damsville Y artersville K X rooksville M angle bisectors of the triangle formed b the three roads and locate the fountain there. Finding engths of Medians M is the centroid of W, and WM. Find WX. he centroid is the point of concurrenc of the medians Z of a triangle. M he medians of a triangle are concurrent at a point that is X two-thirds the distance from each verte to the midpoint of the opposite side. (heorem -) ecause M is the centroid of W, WM WX. WM WX heorem - WX ubstitute for WM. WX Multipl each side b. ail otetaking Guide Geometr esson - ame lass ate esson - esson bjectives Write the negation of a statement and the inverse and contrapositive of a conditional statement Use indirect reasoning Vocabular and Ke oncepts. 00 trand: Geometr opics: Mathematical easoning egation, Inverse, and ontrapositive tatements tatement ample mbolic Form You ead It onditional If an angle is a straight angle, p q If p, then q. then its measure is 0. egation (of p) n angle is not a straight angle. p ot p. Inverse If an angle is not a straight angle, p q If not p, then not q. then its measure is not 0. ontrapositive If an angle s measure is not 0, q p If not q, then not p. then it is not a straight angle. Writing an Indirect roof tep tate as an assumption the opposite (negation) of what ou want to prove. tep how that this assumption leads to a contradiction. tep onclude that the assumption must be false and that what ou want to prove must be true. he negation of a statement has the opposite truth value from that of the original statement. he inverse of a conditional statement negates both the hpothesis and the conclusion. he contrapositive of a conditional switches the hpothesis and the conclusion and negates both. conditional and its contrapositive alwas have the same truth value. quivalent statements have the same truth value. In indirect reasoning all possibilities are considered and then all but one are proved false. he remaining possibilit must be true. n indirect proof is a proof involving indirect reasoning. W Y Inverses, ontrapositives, and Indirect easoning 00 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 ll-in-ne nswers Version Geometr

128 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. Writing the egation of a tatement Write the negation of is not a conve polgon. he negation of a statement has the opposite truth value. he negation of is not in the original statement removes the word not. he negation of is not a conve polgon is is a conve polgon. Identifing ontradictions Identif the two statements that contradict each other. I., Q, and are coplanar. II., Q, and are collinear. III. m Q 0 wo statements contradict each other when the cannot both be true at the same time. amine each pair of statements to see whether the contradict each other. I and II I and III II and III Writing the Inverse and ontrapositive Write the inverse and contrapositive of the conditional statement If is equilateral, then it is isosceles. o write the inverse of a conditional, negate both the hpothesis and the conclusion. Hpothesis onclusion onditional: If is equilateral, then it is isosceles. egate both. Inverse: If is not equilateral, then it is not isosceles. o write the contrapositive of a conditional, switch the hpothesis and conclusion, then negate both. Hpothesis onclusion onditional: If is equilateral, then it is isosceles. witch and negate both. he First tep of an Indirect roof Write the first step of an indirect proof. rove: triangle cannot contain two right angles. In the first step of an indirect proof, ou assume as true the negation of what ou want to prove. ecause ou want to prove that a triangle cannot contain two right angles, ou assume that a triangle can contain two right angles. he first step is ssume that a triangle contains two right angles. 0 ontrapositive: If is not isosceles, then it is not equilateral. Geometr esson - ail otetaking Guide ame lass ate. Write (a) the inverse and (b) the contrapositive of Maa ngelou s statement If ou don t stand for something, ou ll fall for anthing. a. If ou stand for something, ou won t fall for anthing. b. If ou won t fall for anthing, then ou stand for something.. You want to prove each statement true. Write the first step of an indirect proof. a. he shoes cost no more than $0. he shoes cost more than $0. b. m m ml # ml. Identif the two statements that contradict each other. plain. I. FGK II. FG ' K III. FG > K I and II; wo segments cannot be parallel as well as perpendicular. I and II contradict each other. wo segments can be parallel and congruent. I and III do not contradict each other. wo segments can be perpendicular and congruent. II and III do not contradict each other.. ritical hinking You plan to write an indirect proof showing that X is an obtuse angle. In the first step ou assume that X is an acute angle. What have ou overlooked?, Q, and are coplanar and collinear. hree points that lie on the same line are both coplanar and collinear, so these two statements do not contradict each other., Q, and are coplanar, and m Q 0. hree points that lie on an angle are coplanar, so these two statements do not contradict each other. Indirect roof Write an indirect proof. rove: cannot contain two obtuse angles., Q, and are collinear, and m Q 0. If three distinct points are collinear, the form a straight angle, so m Q cannot equal 0. tatements II and III do contradict each other. tep ssume as true the opposite of what ou want to prove. hat is, assume that contains two obtuse angles. et l and l be obtuse. tep If and are obtuse, m m m 0. his contradicts the riangle ngle-um heorem, which states that m m m 0. tep he assumption in tep must be false contain two obtuse angles. Quick heck.. Write the negation of each statement. a. m XYZ 0. he measure of XYZ is not more than 0. b. oda is not uesda. oda is uesda.. cannot ail otetaking Guide Geometr esson - ame lass ate esson - esson bjectives Use inequalities involving angles of triangles Use inequalities involving sides of triangles Ke oncepts. omparison ropert of Inequalit If a b c and c 0, then a b. 00 trand: Geometr opics: elationships mong Geometric Figures orollar to the riangle terior ngle heorem he measure of an eterior angle of a triangle is greater than the measure of each of its remote interior angles. m m and m m heorem -0 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. If XZ XY, then m Y m Z. heorem - If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. If m m, then. heorem -: riangle Inequalit heorem he sum of the lengths of an two sides of a triangle is greater than the length of the third side. M M M M M M 0 Inequalities in riangles X M Y Z X could be a right angle. 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 Geometr ll-in-ne nswers Version

129 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. ppling the orollar plain wh m m. 0 is an eterior Geometr esson - angle of XY. he orollar to the terior ngle heorem states that the measure of an eterior angle of a triangle is greater than the measure of each of its remote interior angles. he remote interior angles are and, so m m. and are corresponding angles formed b transversal. ecause XY, ou can conclude that b the orresponding ngles ostulate. hus, m m. ubstituting m for m in the inequalit m m produces the inequalit m m. ppling heorem -0 In GY, G, GY, and Y 0. ist the angles from largest to smallest. heorem -0: If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. o two sides of GY are congruent, so the largest angle lies opposite the longest side. Y Find the angle opposite each side. he longest side is 0. he opposite angle, G, is the largest. he shortest side is. he opposite angle,, is the smallest. From largest to smallest, the angles are G, Y,. Using the riangle Inequalit heorem an a triangle have sides with the given lengths? plain. a. cm, cm, cm ccording to the riangle Inequalit heorem, the sum of the lengths of an two sides of a triangle is greater than the length of the third side. he sum of and is not greater than. o cannot, a triangle have sides with these lengths. b. in., in., in. Yes can, a triangle have sides with these lengths. Y X G 0 ail otetaking Guide ame lass ate esson - esson bjective efine and classif special tpes of quadrilaterals Ke oncepts. pecial Quadrilaterals Kite o pairs of parallel sides Isosceles rapezoid parallelogram is a quadrilateral with both pairs of opposite sides parallel. rhombus is a parallelogram with four congruent sides. Quadrilateral pair of parallel sides rapezoid rectangle is a parallelogram with four right angles. trapezoid is a quadrilateral with eactl one pair of parallel sides. 00 trand: Geometr opic: elationships mong Geometric Figures pairs of parallel sides ectangle kite is a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent. square is a parallelogram with four congruent sides and four right angles. n isosceles trapezoid is a trapezoid whose nonparallel sides are congruent. lassifing Quadrilaterals arallelogram quare hombus Finding ossible ide engths In FGH, FG m and GH m. escribe the possible lengths of FH. he riangle Inequalit heorem: he sum of the lengths of an two sides of a triangle is greater than the length of the third side. olve three inequalities. FH FH FH FH FH FH FH must be longer than m and shorter than m. Quick heck.. Use the diagram in ample. If Y XY, eplain wh m m. m m b the Isosceles riangle heorem m m b the orollar to the riangle terior ngle heorem m m b ubstitution. ist the angles of in order from smallest to largest.,,. an a triangle have sides with the given lengths? plain. a. m, m, and m no; is not greater than b. d, d, and d es;,, and. triangle has sides of lengths in. and in. escribe the possible lengths of the third side. ail otetaking Guide Geometr esson - ame lass ate amples. lassifing b oordinate Methods etermine the most precise name for the quadrilateral with vertices Q(, ), (, ), H(, ), and (0, ). Graph quadrilateral QH. Find the slope of each side. H lope of Q ( ) lope of H ( ) Q lope of H 0 lope of Q 0 H is parallel to Q because their slopes are equal. H is not parallel to Q because their slopes are not equal. ne pair of opposite sides is parallel, so QH is a trapezoid. et, use the distance formula to see whether an pairs of sides are congruent. Q ( ( )) ( ) " H (0 ) ( ) " H ( ( )) ( ) Q ( 0 ) ( ) 0 ft ecause Q H, QH is an isosceles trapezoid. Using the roperties of pecial Quadrilaterals In parallelogram U, m 0 and m 0. Find. U is a parallelogram. U m m 0 Given ft ft U ( 0) ( 0) efinition of parallelogram If lines are parallel, then interior angles on the same side of a transversal are supplementar. ( 0) ( 0) ubstitute for m and for m. 0 0 implif. 0 ubtract 0 from each side. ivide each side b Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 ll-in-ne nswers Version Geometr

130 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Quick heck.. a. Graph quadrilateral with vertices (, ), (, ), (, ), and (, ). b. lassif in as man was as possible. is a quadrilateral because it has four sides. It is a parallelogram because both pairs of opposite sides are parallel. It is a rhombus because all four sides are congruent. It is a rectangle because it has four right angles and its opposite sides are congruent. It is a square because it has four right angles and its sides are all congruent. esson - esson bjectives Use relationships among sides and among angles of parallelograms Use relationships involving diagonals of parallelograms and transversals Vocabular and Ke oncepts. roperties of arallelograms 00 trand: Geometr opic: elationships mong Geometric Figures 0 c. Which name gives the most information about? plain. quare; the name square means a figure is both a rectangle and a rhombus, since its sides are all congruent and it has four right angles.. Find the values of the variables in the rhombus. hen find the lengths of the sides. a, b, Geometr esson - a b b a ail otetaking Guide ame lass ate Using lgebra Find the value of in. hen find m. pposite angles of a parallelogram are congruent dd to each side. 0 ubtract from each side. 0 ivide each side b. m 0 ubstitute 0 for. m m 0 onsecutive angles of a parallelogram are supplementar. m 0 ubstitute for m. 0 m ubtract from each side. Using lgebra Find the values of and in KM. K M ( ) istribute. implif. he diagonals of a parallelogram bisect each other. ubstitute for in the second equation to solve for. ubtract from each side. ivide each side b. ( ubstitute for in the first equation ) to solve for. o and. implif.. ( ) ( ) heorem - pposite sides of a parallelogram are congruent. heorem - pposite angles of a parallelogram are congruent. heorem - he diagonals of a parallelogram bisect each other. heorem - If three (or more) parallel lines cut off congruent segments on one transversal, then the cut off congruent segments on ever transversal. F onsecutive angles of a polgon share a common side. amples J K M In JKM, J and M are consecutive angles, as are J and K. J and are not consecutive. F ail otetaking Guide Geometr esson - ame lass ate off congruent segments on one transversal, then the cut off congruent segments on ever transversal. plain how to divide a blank card into five equal rows using heorem - and a sheet of lined paper. lace a corner of the top of the card on the first line of the lined paper. lace a corner that forms a consecutive angle with the first corner on the sith line. Mark the points where the lines intersect one side of the card. Mark the points where the lines intersect the opposite side of the card. onnect the marks on opposite sides using a straightedge. Quick heck.. Find the value of in FGH. hen find m, m F, m G, and m H. ; ml 0, mlf 0, mlg 0, and mlh 0. Find the values of a and b. a, b. In the figure, H G F,, and F.. Find H.. W U F G ( ) ( ) H X b 0 a a b F. G Z H Y Using heorem - heorem - states If three (or more) parallel lines cut Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

131 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective etermine whether a quadrilateral is a parallelogram Ke oncepts. Geometr esson - 00 trand: Geometr opic: Geometr heorem - If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. heorem - If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. heorem - If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. heorem - If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. amples. Finding Values for arallelograms Find values for and for which must be a parallelogram. If the diagonals of quadrilateral bisect each other, then is a parallelogram b heorem -. Write and solve two equations to find values of and for which the diagonals bisect each other. 0 iagonals of parallelograms bisect each other. ollect the variables on one side. 0 olve. If and, then is a parallelogram. 0 roving hat a Quadrilateral Is a arallelogram 0 G 0 ail otetaking Guide ame lass ate Quick heck.. Find the values of a and c for which Q must be a parallelogram. a 0, c. an ou prove the quadrilateral is a parallelogram? plain. a. Given: Q >, Q rove: Q is a parallelogram. Yes; a pair of opposite sides are parallel and congruent (heorem -). b. Given: H > GH, H > FH rove: FG is a parallelogram. o; the diagonals do not necessaril bisect each other the figure could be a trapezoid, with G and F being parallel but of unequal length. G Q Q (a 0) a c c H F Is the Quadrilateral a arallelogram? an ou prove the quadrilateral is a parallelogram from what is given? plain. a. Given: > rove: is a parallelogram. ll ou know about the quadrilateral is that onl one pair of opposite sides is congruent. herefore, ou cannot conclude that the quadrilateral is a parallelogram. b. Given: m m G, m F 0 rove: FGH is a parallelogram. he sum of the measures of the angles of a polgon is F 0 (n )0 where n represents the number of sides, so the sum of the measures of the angles of a quadrilateral is ( )0 0. If represents the measure of the unmarked angle, 0 0, so = 0. heorem - states If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ecause both pairs of opposite angles are congruent, the quadrilateral is a parallelogram b heorem -. ail otetaking Guide Geometr esson - G ame lass ate esson - esson bjectives Use properties of diagonals of rhombuses and rectangles etermine whether a parallelogram is a rhombus or a rectangle Ke oncepts. 00 trand: Geometr opic: Geometr hombuses heorem - ach diagonal of a rhombus bisects two angles of the rhombus. bisects, so bisects, so heorem -0 he diagonals of a rhombus are perpendicular. ' ectangles heorem - he diagonals of a rectangle are congruent. H arallelograms heorem - If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. heorem - If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. heorem - pecial arallelograms If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

132 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. ample. Finding ngle Measures Find the measures of the numbered angles in the rhombus. heorem - states that each diagonal of a rhombus bisects two angles of the rhombus, so m. heorem -0 states that the diagonals of the rhombus are perpendicular, so m 0. Identifing pecial arallelograms he diagonals of are such that cm and cm. an ou conclude that is a rhombus or a rectangle? plain. cannot be a rectangle, because and the diagonals of a rectangle are congruent (heorem -). ma be a rhombus, but it ma also not be one, depending on whether ' (heorem -0). ecause the four angles formed b the diagonals all must have measure 0, and must be complementar. ecause m, m 0. Finall, because, the Isosceles riangle heorem allows ou to conclude that. o m. Finding iagonal ength ne diagonal of a rectangle has length. he other diagonal has length. Find the length of each diagonal. heorem -, the diagonals of a rectangle are congruent. iagonals of a rectangle are congruent. ubtract from each side. ubtract from each side. ivide each side b. ( ) ubstitute. ( ) ubstitute. he length of each diagonal is. Quick heck.. Find the measures of the numbered angles in the rhombus. m 0, m 0, m 0, m 0 Geometr esson - 0 ail otetaking Guide ame lass ate esson - esson bjective Verif and use properties of trapezoids and kites Vocabular and Ke oncepts. 00 trand: Geometr opic: elationships mong Geometric Figures rapezoids heorem - he base angles of an isosceles trapezoid are congruent. heorem - he diagonals of an isosceles trapezoid are congruent. he base angles of a trapezoid are two angles that share a base of the trapezoid. eg Kites heorem - he diagonals of a kite are perpendicular. ase angles eg ase angles rapezoids and Kites Quick heck.. Find the length of the diagonals of rectangle GF if F and G =.. parallelogram has angles of 0, 0, 0, and 0. an ou conclude that it is a rhombus or a rectangle? plain. o; there is not enough information to conclude that the parallelogram is a rhombus. It cannot be a rectangle because it has no right angles. ail otetaking Guide Geometr esson - ame lass ate amples. Finding ngle Measures in rapezoids WXYZ is an isosceles trapezoid, and m X. Find m Y, m Z, and m W. W wo angles that share m X m W 0 a leg of a trapezoid are supplementar. m W 0 ubstitute. m W ubtract from each side. ecause the base angles of an isosceles trapezoid are congruent, m Y m X and m Z m W. Using Isosceles rapezoids Half of a spider s web is shown at the right, formed b laers of congruent isosceles trapezoids. Find the measures of the angles in. rapezoid is part of an isosceles triangle whose verte is at the center of the web. ecause there are adjacent congruent verte angles at the center of the web, together forming a straight angle, each verte angle measures 0, or 0. the riangle ngle-um heorem, m m 0 0, so m m 0. ecause is part of an isosceles triangle, m m, so (m ) 0 and m m. nother wa to find the measure of each acute angle is to divide the difference between 0 and the measure of the verte angle b : 0 0. ecause the bases of a trapezoid are parallel, the two angles that share a leg are supplementar, so m m 0 = 0. Quick heck.. In the isosceles trapezoid, m 0. Find m, m Q, and m. 0, 0, 0 F G X Y Q Z 0 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 Geometr ll-in-ne nswers Version

133 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate ample. Finding ngle Measures in Kites Find m, m, and m in the kite. U perpendicular m 0 iagonals of a kite are. U efinition of a kite m Isosceles riangle heorem m m U 0 riangle ngle-um heorem m U 0 iagonals of a kite are perpendicular. 0 0 m ubstitute. 0 m implif. m ubtract from each side. Quick heck.. he middle ring of the piece of ceiling shown is made from congruent isosceles trapezoids. Imagine a circular glass ceiling made from the ceiling pieces with angles meeting at the center. What are the measures of the two sets of base angles of the trapezoids in the middle ring? he measure of both inner angles is. he measure of both outer angles is.. Find m, m, and m in the kite. m 0, m, and m Geometr esson - ail otetaking Guide ame lass ate ample. aming oordinates Use the properties of parallelogram to find the missing coordinates. o not use an new variables. he verte is the origin with coordinates (0, 0). ecause point is p units to the left of point, point is also p units to the left of point, because is a parallelogram. o the first coordinate of point is p. ecause and is horizontal, is also horizontal. o point has the same second coordinate, q, as point. he missing coordinates are (0, 0) and ( p, q). Quick heck.. Use the properties of parallelogram Q to find the missing coordinates. o not use an new variables. Q(s b, c) (, 0) ( p, q) (b, c) Q(?,?) (s, 0) esson - esson bjective ame coordinates of special figures b using their properties ample. roving ongruenc how that WVU is a parallelogram b proving pairs of opposite sides congruent. W(a c, b d) If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a (a, b) V(c e, d) parallelogram b heorem -. You can prove that WVU is a parallelogram b showing that U(e, 0) W VU and WV U. Use the distance formula. a c b d c e a b d e 0 Use the coordinates (, ), W(, ), V(, ), and U(, ). ( ) ( ) 00 trand: Geometr opic: osition and irection W a c a b d b c d VU ( c e e ) ( d 0 ) c d WV ( a c c e ) ( b d d ) (a e) b U ( a e ) ( b 0 ) (a e) b parallelogram ecause W VU and WV U, WVU is a. Quick heck.. Use the diagram above. Use a different method: how that WVU is a parallelogram b finding the midpoints of the diagonals. Midpoint of V a a c e, midpoint of UW. hus, the b d b diagonals bisect each other, and WVU is a parallelogram. lacing Figures in the oordinate lane ail otetaking Guide Geometr esson - ame lass ate esson - esson bjective rove theorems using figures in the coordinate plane Vocabular and Ke oncepts. 00 trand: Geometr opics: osition and irection; Mathematical easoning heorem -: rapezoid Midsegment heorem () he midsegment of a trapezoid is parallel to the bases. () he length of the midsegment of a trapezoid is half the sum of the lengths of the bases. he midsegment of a trapezoid is the segment that joins the midpoints of the nonparallel opposite sides of the trapezoid. M M, M, and M ( ) amples. lanning a oordinate Geometr roof amine trapezoid. plain wh ou can assign the same -coordinate to points and. In a trapezoid, onl one pair of sides is parallel. In,. ecause lies on the horizontal -ais, also must be horizontal. he -coordinates of all points on a horizontal line are the same, so points and have the same -coordinates. roofs Using oordinate Geometr (b, c) (d, c) M (a, 0) Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

134 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Using oordinate Geometr Use coordinate geometr to prove that the quadrilateral formed b connecting the midpoints of rhombus is a rectangle. raw quadrilateral XYZW b connecting the midpoints of. Z( a, b) ( a, 0) Y( a, b) From esson -, ou know that XYZW is a parallelogram. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle from heorem. o show that XYZW is a rectangle, find the lengths of its diagonals, and then compare them to show that the are equal. XZ ( a a) (b ( b)) ( a ) ( b ) a b ( YW ( a a) ( b b) a ) ( b ) a b XZ YW ecause the diagonals are congruent, parallelogram XYZW is a rectangle. Geometr esson - esson - esson bjective Write ratios and solve proportions Vocabular and Ke oncepts. (0, b) W(a, b) (a, 0) X(a, b) (0, b) 00 trand: Geometr opic: osition and irection roperties of roportions a c b d is equivalent to () ad bc () b d () a b () a b a c c d b ail otetaking Guide ame lass ate proportion is a statement that two ratios are equal. a c b d and a : b c : d are eamples of proportions. n etended proportion is a statement that three or more ratios are equal. is an eample of an etended proportion. he ross-roduct ropert states that the product of the etremes of a proportion is equal to the product of the means. means a c b d a : b c : d a d b c etremes scale drawing is a drawing in which all lengths are proportional to corresponding actual lengths. scale is the ratio of an length in a scale drawing to the corresponding actual length. he lengths ma be in different units. amples. Finding atios scale model of a car is in. long. he actual car is ft long. What is the ratio of the length of the model to the length of the car? Write both measurements in the same units. ft in. 0 in. length of model length of car in. ft 0 in. in. 0 atios and roportions c d d Quick heck.. omplete the proof of heorem -. Given: M is the midsegment of trapezoid. (b, c) (d, c) rove: M, M, and M = ( + ) M a. Find the coordinates of midpoints M and. How do the multiples of help? (a, 0) M(b, c) and (a d, c); starting with multiples of eliminates fractions when using the midpoint formula. b. Find and compare the slopes of M,, and. 0, 0, 0; the are all parallel. c. Find and compare the lengths M,, and. M d a b, a, d b. o d a b which is twice M. o the midsegment is half the sum of the bases. d. In parts (b) and (c), how does placing a base along the -ais help? he base along the -ais allows calculating length b subtracting -values.. Use the diagram from ample. plain wh the proof using (a, 0), (0, b), ( a, 0), and (0, b) is easier than the proof using (a, 0), (0, b), ( a, 0), and (0, b). Using multiples of in the coordinates for,,, and eliminates the use of fractions when finding midpoints, since finding midpoints requires division b. ail otetaking Guide Geometr esson - ame lass ate roperties of roportions a b omplete: If b, then. ab ross-roduct ropert ab a a ivide each side b a. b a implif. olving for a Variable olve each proportion. a. n n ross-roduct n 0 implif. n ivide each side b. b. ross-roduct istributive ubtract from each side. ( ) ropert ( ) ropert ropert Quick heck.. photo that is in. wide and in. high is enlarged to a poster that is ft wide and ft high. What is the ratio of the height of the photo to the height of the poster? :. Write two epressions that are equivalent to m n. nswers ma var. ample: m n, m n he ratio of the length of the scale model to the length of the car is :. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

135 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate ample. Using roportions wo cities are in. apart on a map with the scale in. 0 mi. Find the actual distance. et d represent the actual distance. map distance (in.) actual distance (mi.) in. 0 mi esson - esson bjectives Identif similar polgons ppl similar polgons imilar olgons 00 trand: Geometr and Measurement opics: ransformation of hapes and reservation of roperties; Measuring hsical ttributes d d he cities are actuall Quick heck. 0 d 0 ubstitute. ( ) ross-roduct ropert implif. miles apart.. olve each proportion. a. b. n n z 0 z 0. n. ecall ample. You want to make a new map with a scale of in. mi. wo cities that are actuall miles apart are to be represented on our map. What would be the distance in inches between the cities on the new map? 0 inches Geometr esson - ail otetaking Guide ame lass ate amples. etermining imilarit etermine whether the K parallelograms are similar. plain. 00 heck that corresponding sides are proportional. JK 0 K MJ M orresponding sides of the two parallelograms are proportional. heck that corresponding angles are congruent. corresponds to K, but m m K, so corresponding J M angles are not congruent. lthough corresponding sides are proportional, the parallelograms are not similar because the corresponding angles are not congruent. Using imilar Figures If YXZ, find the value of. Y ecause YXZ, ou can write and solve a proportion. 0 0 orresponding sides are YZ X Z XZ proportional ubstitute. olve for. implif. Using the Golden atio he dimensions of a rectangular tabletop are in the Golden atio. he shorter side is 0 in. Find the longer side. et represent the longer side of the tabletop Write a proportion using the Golden atio. Vocabular. imilar figures have the same shape but not necessaril the same size. wo polgons are he mathematical smbol for similarit is. he similarit ratio is the ratio of the lengths of corresponding sides of similar figures. golden rectangle is a rectangle that can be divided into a square and a rectangle that is he golden ratio is the ratio of the length to the width of an golden rectangle, about ample. similar if corresponding angles are congruent and corresponding sides are proportional. similar to the original rectangle.. :. Understanding imilarit XYZ. omplete each statement. a. m Y and m Y, so m because congruent angles have the same measure. b. YZ XZ Quick heck. corresponds to, so XZ YZ. XZ. efer to the diagram for ample. omplete: m and YZ XY ail otetaking Guide Geometr esson - ame lass ate Quick heck.. ketch XYZ and M with X M, Y, and Z. lso, XY, YZ, ZX, M,, and M. an ou conclude that the two triangles are similar? Z Y X. efer to the diagram for ample. Find. Yes; corresponding angles are congruent and corresponding sides are proportional.. golden rectangle has shorter sides of length 0 cm. Find the length of the longer sides. about. cm M X Y Z. ross-roduct ropert he table is about in. long. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

136 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Use,, and similarit statements ppl,, and similarit statements Vocabular and Ke oncepts. ostulate -: ngle-ngle imilarit ( ) ostulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. M heorem -: ide-ngle-ide imilarit ( ) heorem If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar. heorem -: ide-ide-ide imilarit ( ) heorem If the corresponding sides of two triangles are proportional, then the triangles are similar. Indirect measurement is a wa of measuring things that are difficult to measure directl. amples. Using the ostulate MX '. plain wh the triangles are similar. Write a similarit statement. ecause MX ', XM and XK are right angles, so XM > XK. > because their measures are equal. MX KX b the ngle-ngle imilarit ( ) ostulate. Geometr esson - 00 trand: Geometr opic: ransformation of hapes and reservation of roperties roving riangles imilar M X K M ail otetaking Guide ame lass ate Quick heck.. In ample, ou have enough information to write a similarit statement. o ou have enough information to find the similarit ratio? plain. o; none of the side lengths are given.. plain wh the triangles at the right must be similar. Write a similarit statement. G GF F, so the triangles are similar b the ~ heorem; n, nfg. In sunlight, a cactus casts a -ft shadow. t the same time, a person ft tall casts a -ft shadow. Use similar triangles to find the height of the cactus. ft. ft ft ft ft G F Using imilarit heorems plain wh the triangles must be similar. Write a similarit statement. YVZ > WVX because the are vertical angles. VY and VZ VW VX, so corresponding sides are proportional. herefore, YVZ WVX b the ide-ngle-ide imilarit ( ) heorem. Using imilar riangles Joan places a mirror ft from the base of a tree. When she stands ft from the mirror, she can see the top of the tree reflected in it. If her ees are ft above the ground, how tall is the tree? represents the height of the tree, oint M represents the mirror, and point J represents Joan s ees. oth Joan and the tree are perpendicular to the ground, so m JM m M, and therefore ljm lm. he light reflects off the mirror at the same angle at which it hits the mirror, so JM M. Use similar triangles to find the height of the tree. njm, nm J he tree is M M 0 0 feet tall. orresponding ubstitute. olve for. ostulate sides of similar triangles are proportional. Z V Y J W M ail otetaking Guide Geometr esson - ame lass ate esson - esson bjective Find and use relationships in similar right triangles Vocabular and Ke oncepts. heorem - he altitude to the hpotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and similar to each other. orollar to heorem - he length of the altitude to the hpotenuse of a right triangle is the geometric mean of the lengths of the segments of the hpotenuse. orollar to heorem - he altitude to the hpotenuse of a right triangle separates the hpotenuse so that the length of each leg of the triangle is the geometric mean of the length of the adjacent hpotenuse segment and the length of the hpotenuse. he geometric mean of two positive numbers a and b is the positive number such that a b. amples. Finding the Geometric Mean Find the geometric mean of and. Write a proportion. ross-roduct ropert. 00 trand: Geometr opic: ransformation of hapes and reservation of roperties X imilarit in ight riangles! Find the positive square root. he geometric mean of and is. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

137 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Finding istance t a golf course, Maria drove her ball d straight toward the cup. Her brother Gabriel drove his ball straight 0 d, but not toward the cup. he diagram shows the results. Find and, their remaining distances from the cup. Use orollar of heorem - to solve for. 0 Write a proportion. 0 G up M 0 esson - esson bjectives Use the ide-plitter heorem Use the riangle-ngle-isector heorem 00 trand: Geometr roportions in riangles opic: ransformation of hapes and reservation of roperties ( ) 0 ross-roduct ropert,,00 istributive ropert 0, olve for. 0 Use orollar of heorem - to solve for. Write a proportion ,00!,00 0 Geometr esson - ubstitute 0 for. implif. ross-roduct Find the positive square root. ropert Maria s ball is 0 d from the cup, and Gabriel s ball is 0 d from the cup. Quick heck.. Find the geometric mean of and 0.. ecall ample. Find the distance between Maria s ball and Gabriel s ball. 0" d ail otetaking Guide ame lass ate Using the orollar to the ide-plitter heorem he segments joining the sides of trapezoid U are parallel to its bases. Find and. ecause U is a trapezoid, U. ross-roduct olve for.... ubstitute for.. ross-roducts. olve for. and. Quick heck.. Use the ide-plitter heorem to find the value of... olve for and. ;. orollar to the ide-plitter orollar to the ide-plitter heorem ropert heorem ropert U Ke oncepts. heorem -: ide-plitter heorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionall. heorem -: riangle-ngle-isector heorem If a ra bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. orollar to heorem - If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. amples. Using the ide-plitter heorem Find. 0 M M 0. a b ubstitute. olve for. c d ide-plitter ross-roduct heorem ropert ail otetaking Guide Geometr esson - a b c d M ame lass ate ample. Using the riangle-ngle-isector heorem Find the value of. G I 0 K Quick heck. 0 0 IG GH 0. Find the value of.. IK HK 0 ( 0 ) H ubstitute. olve for. riangle-ngle-isector heorem Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

138 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Use the thagorean heorem Use the onverse of the thagorean heorem he thagorean heorem and Its onverse 00 trand: Geometr opic: elationships mong Geometric Figures ample. thagorean riples Find the length of the hpotenuse of #. o the lengths of the sides of # form a thagorean triple? a b c Use the thagorean heorem. c ubstitute for a, and for b. Vocabular and Ke oncepts. c implif. heorem -: thagorean heorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hpotenuse. a b c heorem -: onverse of the thagorean heorem If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. heorem - If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, the triangle a c is obtuse. b If c a b, the triangle is obtuse. heorem - If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, the triangle a c is acute. b If c a b, the triangle is acute. thagorean triple is a set of nonzero whole numbers a, b, and c that satisf the equation a b c. Geometr esson - c b ail otetaking Guide ame lass ate ample. Using implest adical Form Find the value of b. eave our answer in simplest radical form a b c Use the thagorean b ubstitute for a, for c. b implif. 0 b ubract from both sides. b Î 0 Î b! b ( ) implif. ake the square root. heorem. Quick heck.. he hpotenuse of a right triangle has length. ne leg has length. Find the length of the other leg. eave our answer in simplest radical form.! b a Î c c ake the square root. implif. he length of the hpotenuse is. he lengths of the sides,,, and, form a thagorean triple because the are whole numbers that satisf a b c. Quick heck.. right triangle has a hpotenuse of length and a leg of length 0. Find the length of the other leg. o the lengths of the sides form a thagorean triple?!; no ail otetaking Guide Geometr esson - ame lass ate amples. Is It a ight riangle? Is this triangle a right triangle? a b 0 c 0 ubstitute for a, for b, and for c. 0 implif. ecause a b c, the triangle is not a right triangle. lassifing riangles as cute, btuse, or ight he numbers 0,, and 0 represent the lengths of the sides of a triangle. lassif the triangle as acute, obtuse, or right. c 0 a b ompare c with a b ubstitute the greatest length for c implif. 00. ecause c a b, the triangle is a(n) obtuse triangle. Quick heck.. triangle has sides of lengths,, and 0. Is the triangle a right triangle? no. triangle has sides of lengths,, and. lassif the triangle b its angles. acute m m m Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

139 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Use the properties of - -0 triangles Use the properties of triangles pecial ight riangles 00 trand: Geometr opic: elationships mong Geometric Figures ppling the - -0 riangle heorem he distance from one corner to the opposite corner of a square plaground is ft. o the nearest foot, how long is each side of the plaground? he distance from one corner to the opposite corner, ft, is the length of the hpotenuse of a - -0 triangle. "? leg hpotenuse "? leg s s ft Ke oncepts. heorem -: - -0 riangle heorem In a - -0 triangle, both legs are congruent and the length of the hpotenuse is " times the length of a leg. hpotenuse "? leg heorem -: riangle heorem In a triangle, the length of the hpotenuse is twice the length of the shorter leg. he length of the longer leg is " times the length of the shorter leg. hpotenuse? shorter leg longer leg "? shorter leg amples. Finding the ength of the Hpotenuse Find the value of the variable. Use the - -0 riangle heorem to find the hpotenuse. h "?! hpotenuse "? leg h " implif. h ( ) h ( ) h 0 " he length of the hpotenuse is 0". Geometr esson - h s s s s 0 s 0 s ail otetaking Guide ame lass ate ample. ppling the riangle heorem rhombus-shaped garden has a perimeter of 00 ft and a 0 angle. Find the area of the garden to the nearest square foot. ft h ecause a rhombus has four sides of equal length, each side is ft. 0 ecause a rhombus is also a parallelogram, ou can use the area formula bh. raw the altitude h, and then solve for h. he height h is the longer leg of the right triangle. o find the height h, ou can use the properties of triangles. hen appl the area formula.? shorter leg hpotenuse? shorter leg. shorter leg ivide each side b. h. " longer leg "? shorter leg parallelogram bh Use the formula for the area of a. ( ) (.! ) ubstitute for b and." for h.. Use a calculator. o the nearest square foot, the area is ft. Quick heck.. Find the value of each variable.,!. rhombus has 0-in. sides, two of which meet to form the indicated angle. Find the area of each rhombus. (Hint: Use a special right triangle to find height.) a. a 0 angle b. a 0 angle 0 in. 0! in. 0 0 leg ivide each side b ". " leg. Use a calculator. ach side of the plaground is about ft. Using the ength of ne ide Find the value of each variable. Use the riangle heorem to find the lengths of the legs.!? hpotenuse? shorter leg 0 " ivide each side b. " implif. 0 "? shorter leg riangle heorem "?! ubstitute " for shorter leg.? "? " implif. he length of the shorter leg is ", and the length of the longer leg is. Quick heck.. Find the length of the hpotenuse of a - -0 triangle with legs of length!.!. square garden has sides of 00 ft long. You want to build a brick path along a diagonal of the square. How long will the path be? ound our answer to the nearest foot. ft ail otetaking Guide Geometr esson - ame lass ate esson - esson bjective Use tangent ratios to determine side lengths in triangles Vocabular. he tangent of acute in a right triangle is the ratio of the length of the leg opposite l to the length of the leg adjacent to l. eg opposite tangent of opposite You can abbreviate the equation as tan adjacent. amples. Writing angent atios Write the tangent ratios for and. 0 eg adjacent to opposite tan adjacent opposite tan adjacent 00 trand: Measurement opic: Measuring hsical ttributes length of leg opposite l length of leg adjacent to l he angent atio 0 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

140 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Using a angent atio o measure the height of a tree, lma walked ft from the tree and measured a angle from the ground to the top of the tree. stimate the height of the tree. Quick heck.. a. Write the tangent ratios for K and J. J, ft K he tree forms a right tangent ratio to estimate the height of the tree. angle with the ground, so ou can use the tan Use the tangent ratio. height ( tan ) olve for height..0 Use a calculator. he tree is about feet tall. Using the Inverse of angent Find m to the nearest degree. 0 o m. tan ( m tan )...00 Geometr esson - height ail otetaking Guide ame lass ate esson - esson bjective Use sine and cosine to determine side lengths in triangles Vocabular. he sine of is the ratio of the length of the leg opposite to the length of the hpotenuse. he cosine of is the ratio of the length of the leg adjacent to to the hpotenuse. sine of opposite his can be abbreviated sin hpotenuse. cosine of adjacent his can be abbreviated cos hpotenuse. n identit is an equation that is true for all allowed values of the variable. n eample of a trigonometric identit is sin cos (0 ). amples. leg opposite l hpotenuse leg adjacent to l hpotenuse 00 trand: Measurement opic: Measuring hsical ttributes hpotenuse eg adjacent to Writing ine and osine atios Use the triangle to find sin, cos, sin G, and cos G. Write our answers in simplest terms. opposite sin hpotenuse 0 adjacent cos hpotenuse 0 opposite sin G hpotenuse 0 adjacent cos G hpotenuse 0 ine and osine atios eg opposite 0 G b. How is tan K related to tan J? he are reciprocals.. Find the value of w to the nearest tenth. a. 0 b. c. w.0. w.... Find m Y to the nearest degree. ail otetaking Guide Geometr esson - 00 w ame lass ate Using the osine atio 0-ft wire supporting a flagpole forms a angle with the flagpole. o the nearest foot, how high is the flagpole? he flagpole, wire, and ground form a right triangle with the wire as the hpotenuse. ecause ou know an angle and the measure of its hpotenuse, ou can use the cosine ratio to find the height of the flagpole. height cos Use the cosine ratio. 0 height 0? cos olve for height. 0.0 Use a calculator. he flagpole is about feet tall. Using the Inverse of osine and ine right triangle has a leg. units long and hpotenuse.0 units long. Find the measures of its acute angles to the nearest degree...0 Use the inverse of the cosine function to find m.. cos 0. Use the cosine ratio..0 cos ( 0. ) Use the inverse of the cosine. m 0.. Use a calculator. m ound to the nearest degree. o find m, use the fact that the acute angles of a right triangle are complementar. m m 0 efinition of complementar angles m m 0 ubstitute. implif. he acute angles, rounded to the nearest degree, measure and. 0 ft Y Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

141 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Quick heck.. a. Write the sine and cosine ratios for X and Y. sin X, cos X, sin Y, cos Y X 0 esson - esson bjective Use angles of elevation and depression to solve problems 00 trand: Measurement opic: Measuring hsical ttributes ngles of levation and epression Z Y Vocabular. n angle of elevation is the angle formed b a horizontal line and the line of sight to an object above the horizontal line. b. In general, how are sin X and cos Y related? plain. sin X cos Y when lx and ly are complementar.. In ample, suppose that the angle the wire makes with the ground is 0. What is the height of the flagpole to the nearest foot? feet. Find the value of. ound our answer to the nearest degree. a. b. 0. Geometr esson - 0 heodolite ft 0 ft 0 ail otetaking Guide ame lass ate urveing surveor stands 00 ft from a building to measure its height with a -ft tall theodolite. he angle of elevation to the top of the building is. How tall is the building? Use a diagram to represent the situation. o find the height of the building, add the height of the theodolite, which is feet tall. he building is about 0 ft ft, or ft tall. viation n airplane fling 00 ft above the ground begins a descent to land at the airport. How man miles from the airport is the airplane when it starts its descent? (ote: he angle is not drawn to scale.) 00. tan he airplane is about 00 Ground Use the 00? tan 0 sin,00,00 ft ratio. olve for. Use a calculator tangent 00 sin Use the olve for. Use a calculator. ivide b feet to miles. sine 0 miles from the airport when it starts its descent. 00 ft ratio. to convert op of building n angle of depression amples. object below the horizontal line. Identifing ngles of levation and epression escribe and as the relate to the situation shown. is the angle formed b a horizontal line and the line of sight to an ngle of depression Horizontal line Horizontal line ngle of elevation ne side of the angle of depression is a horizontal line. is the angle of depression from the airplane to the building. ne side of the angle of elevation is a horizontal line. is the angle of elevation from the building to the airplane. ail otetaking Guide Geometr esson - ame lass ate Quick heck.. escribe each angle as it relates to the situation in ample. a. b. he angle of depression from the building to the person on the ground he angle of elevation from the person on the ground to the building. You sight a rock climber on a cliff at a angle of elevation. he horizontal ground distance to the cliff is 000 ft. Find the line-of-sight distance to the rock climber. You about ft. n airplane pilot sees a life raft at a angle of depression. he airplane s altitude is km. What is the airplane s surface distance d from the raft? about. km 000 ft ock climber Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

142 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives escribe vectors olve problems that involve vector addition Vocabular and Ke oncepts. dding Vectors For a, and c,, a c k, l. vector is an quantit with magnitude (size) and direction. vector can be represented with an arrow. he magnitude of a vector is its size, or length. he initial point of a vector is the point at which it starts. he terminal point of a vector is the point at which it ends. resultant vector is the sum of other vectors. he magnitude of vector is the distance from its initial point to its terminal point. he ordered pair notation for the vector is k, l. Geometr esson - 00 trand: Geometr opic: osition and irection Vectors ail otetaking Guide ame lass ate dding Vectors Vectors v, and w, are shown at the right. Write the sum of the two vectors as an ordered pair. hen draw s, the sum of v and w. o find the first coordinate of s, add the first coordinates of v and w. o find the second coordinate of s, add the second coordinates of v and w. s k, l k, l k, ( ) l dd the coordinates. k, 0 l implif. raw vector v using the origin as the initial point. raw vector w using the terminal point of v, (, ), as the initial point. raw the resultant vector s from the initial point of v to the terminal point of w. Quick heck.. escribe the vector at the right as an ordered pair. Give the coordinates to the nearest tenth. k.,.l. small airplane lands mi east and mi north of the point from which it took off. escribe the magnitude and the direction of its flight vector. about mi at of. Write the sum of the two vectors k, l and k, l as an ordered pair. (, ) (, ) v w s amples. escribing a Vector escribe M as an ordered pair. Give coordinates to the nearest tenth. Use the sine and cosine ratios to find the values of and. cos 0 Use sine and cosine. sin M 0 ( cos 0 ) olve for the variable. 0( sin 0 ). Use a calculator..00 ecause point M is in the third quadrant, both coordinates are negative.o the nearest tenth, M k.,.l. escribing a Vector irection boat sailed mi east and mi south. he trip can be described b the vector k, l. Use distance and direction to describe this vector a second wa. raw a diagram for the situation. o find the distance sailed, use the istance Formula. W 0 0 d ( ) ( ) istance Formula d implif. d " implif. d ake the square root. o find the direction the boat sails, find the angle that the vector forms with the -ais. tan 0. Use the tangent ratio. tan (0.) Find the angle whose tangent is Use a calculator. he boat sailed miles at about south of east. 0 0 ail otetaking Guide Geometr esson - mi (0, 0) mi (, ) ame lass ate esson - esson bjectives Identif isometries Find translation images of figures Vocabular. transformation of a geometric figure is a change in its position, shape, or size. In a transformation, the preimage is the original image before changes are made. In a transformation, the image is the resulting figure after changes are made. n isometr is a transformation in which the preimage and the image are congruent. translation (slide) is a transformation that maps all points the same distance and in the same direction. composition of transformations is a combination of two or more transformations. amples. Identifing Isometries oes the transformation appear to be an isometr? Image reimage turned he image appears to be the same as the preimage, but. ecause the figures appear to be congruent, the transformation appears to be an isometr. ranslations 00 trand: Geometr opic: ransformation of hapes and reservation of roperties; osition and irection k, l 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 Geometr ll-in-ne nswers Version

143 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate aming Images and orresponding arts In the diagram, XYZ is an image of. a. ame the images of and. ecause corresponding vertices of the preimage and the image are listed in the same order, the image of is Y, and the image of is Z. b. ist all pairs of corresponding sides. Z X Y b. ist all pairs of corresponding sides. I and U; I and U; and. Find the image of for the translation (, ). (, ), (, ), (, 0) ecause corresponding sides of the preimage and the image are listed in the same order, the following pairs are corresponding sides: and XY, and XZ, and YZ. Finding a ranslation Image Find the image of under the translation (, ) (, ). (, ) (, ) Use the rule (, ) (, ) (, 0) (, 0 ) (0, ) ( 0, ) he image of is with (, ), (, ), and (, ). Quick heck.. oes the transformation appear to be an isometr? plain. a. b. reimage reimage Image Yes; the figures appear to be congruent b a flip.. In the diagram, I U. a. ame the images of I and point. lu, Geometr esson - Image Yes; the figures appear to be congruent b a flip and a slide. U ail otetaking Guide ame lass ate Quick heck.. Use the rule (, ) (, ) to find the translation image of M. Graph the image M. M M. efer to ample. ritt now makes a third deliver. tarting from where the second deliver was, he goes blocks north and block west. How man blocks is ritt from the pharmac? He is blocks north and blocks west of the pharmac. I amples. Writing a ule to escribe a ranslation Write a rule to describe the translation. You can use an point on and its image on to describe the translation. Using (, ) and its image (, 0), the horizontal change is ( ), or, and the vertical change is 0, or. he rule is (, ) (, ). dding ranslations ritt rides his biccle blocks north and blocks east of a pharmac to deliver a prescription. hen he rides blocks south and blocks west to make a second deliver. How man blocks is he now from the pharmac? he rule (, ) (, ) represents a ride of blocks north and blocks east. he rule (, ) (, ) represents a ride of blocks south and blocks west. ritt s position after the second deliver is the composition of the two translations. ( 0, 0 ) translates to ( 0, 0 ) or (, ). hen, (, ) translates to (, ) or (, ). ritt is block south and blocks west of the pharmac. ail otetaking Guide Geometr esson - ame lass ate esson - esson bjectives Find reflection images of figures Vocabular. 00 trand: Geometr opics: ransformation of hapes and reservation of roperties eflections reflection in line r is a transformation such that if a point is on line r, then the image of is itself, and if a point is not on line r, then its image is the point such that r is the perpendicular bisector of r. ample. Finding eflection Images If point Q(, ) is reflected across line, what are the coordinates of its reflection image? Q is units to the left of the reflection line, so its image Q is units to the right of the reflection line. he reflection line is the perpendicular bisector of QQr if Q is at (, ). Quick heck.. What are the coordinates of the image of Q if the reflection line is? (, ) ine of reflection r Q = Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

144 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. rawing eflection Images XYZ has vertices X(0, ), Y(, 0), and Z(, ). raw XYZ and its reflection image in the line. First locate vertices X, Y, and Z and draw XYZ in a coordinate plane. ocate points X, Y, and Z such that the line of reflection is the perpendicular bisector of XXr, YYr, and ZZr. raw the reflection image X Y Z. ongruent ngles is the reflection of across. how that and W form congruent angles with line. ecause a reflection is an isometr, and r form congruent angles with line. r and W also form congruent angles with line because vertical angles are congruent. herefore, and W form congruent angles with line b the ransitive ropert. Quick heck.. has vertices (, ), (0, ), and (, ). raw and its reflection image in the line.. In ample, what kind of triangle is? Imagine the image of point W reflected across line. What can ou sa about WW and? isosceles; the are similar b Geometr esson - X ZZ X Y Y W ail otetaking Guide ame lass ate Identifing a otation Image egular heagon F is divided into si equilateral triangles. a. ame the image of for a 0 rotation about M. ecause 0 0, each central angle of F measures 0. 0 counterclockwise rotation about center M moves point across four triangles. he image of point is point. b. ame the image of M for a 0 rotation about F. MF is equilateral, so FM has measure rotation of MF about point F would superimpose FM on F, so the image of M under a 0 rotation about point F is point. Quick heck.. raw the image of for a 0 rotation about point. abel the vertices of the image.. egular pentagon is divided into congruent triangles. ame the image of for a rotation about point X. F M X esson - esson bjective raw and identif rotation images of figures Vocabular. rotation of about a point is a transformation for which the following are true: he image of is itself (that is, ). For an point V, V V and m VV. he center of a regular polgon is a point that is equidistant vertices from the polgon s. amples. rawing a otation Image raw the image of under a 0 rotation about. tep Use a protractor to draw a 0 angle at verte with one side. tep Use a compass to construct r. tep ocate and in a similar manner. hen draw. otations 00 trand: Geometr opic: ransformation of hapes and reservation of roperties 0 ail otetaking Guide Geometr esson - V V ame lass ate.amples. Finding an ngle of otation regular -sided polgon can be formed b stacking congruent square sheets of paper rotated about the same center on top of each other. Find the angle of rotation about M that maps W to. W onsecutive vertices of the three squares form the outline of a regular -sided polgon. 0 0, so each verte of the polgon is a 0 rotation about point M. You must rotate counterclockwise through vertices to map point W to point, so the angle of rotation is? 0 0. ompositions of otations escribe the image of quadrilateral XYZW for a composition of a rotation and then a rotation, both about point X. wo rotations of and about the same point make a total rotation of, or 0. ecause this forms a complete rotation about point X, the image is the preimage XYZW. Quick heck.. In the figure from ample, find the angle of rotation about M that maps to K. 0. raw the image of the kite for a composition rotation of two 0 rotations about point K. K M K Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

145 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective Identif the tpe of smmetr in a figure Vocabular. mmetr 00 trand: Geometr opic: ransformation of hapes and reservation of roperties amples. Identifing ines of mmetr raw all lines of smmetr for the isosceles trapezoid. raw an lines that divide the isosceles trapezoid so that half of the figure is a mirror image of the other half. figure has smmetr if there is an isometr that maps the figure onto itself. 0 figure has reflectional smmetr if there is a reflection that maps the figure onto itself. ine smmetr is the same as reflectional smmetr. ine of figure has rotational smmetr if it is its own image for some rotation of 0 or less. figure has point smmetr if it has 0 rotational smmetr. Geometr esson - mmetr he heart-shaped figure has line ( ) smmetr. reflectional he figure is its own image after one half-turn, so it has rotational smmetr with a 0 angle of rotation. he figure also has point smmetr. ail otetaking Guide ame lass ate Quick heck.. raw a rectangle and all of its lines of smmetr.. a. Judging from appearance, tell whether the figure at the right has rotational smmetr. If so, give the angle of rotation. es; 0 b. oes the figure have point smmetr? es. ell whether the umbrella has rotational smmetr about a line and/or reflectional smmetr. he umbrella has both rotational and reflectional smmetr. here is one line of smmetr. Identifing otational mmetr Judging from appearance, do the letters V and H have rotational smmetr? If so, give an angle of rotation. he letter V does not have rotational smmetr because it must be rotated 0 before it is its own image. he letter H is its own image after one half-turn, so it has rotational smmetr with a 0 angle of rotation. Finding mmetr nut holds a bolt in place. ome nuts have square faces, like the top view shown below. ell whether the nut has rotational smmetr and/or reflectional smmetr. raw all lines of smmetr. he nut has a square outline with a circular opening. he square and circle are concentric. he nut is its own image after one quarter-turn, so it has 0 rotational smmetr. he nut has lines of smmetr. ail otetaking Guide Geometr esson - ame lass ate esson - esson bjective ocate dilation images of figures 00 trand: Geometr opic: ransformation of hapes and reservation of roperties ilations Vocabular. dilation is a transformation with center and scale factor n for which the following are true: he image of is itself ) (that is, ). For an point, is on and n. rqr is the image of Q under a dilation Q with center and scale factor. Q n enlargement is a dilation with a scale factor greater than. reduction is a dilation with a scale factor less than. amples. Finding a cale Factor ircle with -cm diameter and center is a dilation of concentric circle with -cm diameter. escribe the dilation. he circles are concentric, so the dilation has center. ecause the diameter of the dilation image is smaller, the dilation is a reduction. diameter of dilation image cale factor: diameter of preimage he dilation is a reduction with center and scale factor. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

146 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate cale rawings he scale factor on a museum s floor plan is : 00. he length and width of one wing on the drawing are in. and in. Find the actual dimensions of the wing in feet and inches. he floor plan is a reduction of the actual dimensions b a scale factor of 00. Multipl each dimension on the drawing b 00 to find the actual dimensions. hen write the dimensions in feet and inches. in in. ft, in. in in. 00 ft Quick heck.. Quadrilateral J K M is a dilation image of quadrilateral JKM. escribe the dilation. J J K K M M he museum wing measures ft, in. b 00 ft. Graphing ilation Images has vertices (, ), (0, ), and (, ). What are the coordinates of the image of for a dilation with center (0, 0) and scale factor 0.? ( ) 0. ( ) 0. (, ) (0.?, ) he scale factor is, so use 0. 0 (0., 0.) (0, ) (, 0. ) the rule (, ). 0. ( ) 0. (, ) (, ) he vertices of the reduction image of are (.,.), (0, ), and (., ). Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Use a composition of reflections Identif glide reflections Vocabular and Ke oncepts. heorem - translation or rotation is a composition of two reflections. heorem - composition of reflections across two parallel lines is a translation. heorem - composition of reflections across two intersecting lines is a rotation. heorem -: Fundamental heorem of Isometries In a plane, one of two congruent figures can be mapped onto the other b a composition of at most three reflections. heorem -: Isometr lassification heorem here are onl four isometries. he are the following: ompositions of eflections 00 trand: Geometr opic: ransformation of hapes and reservation of roperties eflection ranslation otation Glide reflection glide reflection is the composition of a glide (translation) and a reflection across a line parallel to the direction of translation. he dilation is a reduction with center (0, 0) and scale factor.. he height of a tractor-trailer truck is. m. he scale factor for a model truck is. Find the height of the model to the nearest centimeter. cm. Find the image of ZG for a dilation with center (0, 0) and scale factor. raw the reduction on the grid and give the coordinates of the image s vertices. Z Z G G Vertices: (, 0), Z a, b, and G a, b ail otetaking Guide Geometr esson - ame lass ate amples. omposition of eflections in Intersecting ines he letter is reflected in line and then in line. escribe the resulting rotation. Find the image of through a reflection across line. Find the image of the reflection through another reflection across line. he composition of two reflections across intersecting lines is a rotation. he center of rotation is the point where the lines intersect, and the angle is twice the angle formed b the intersecting lines. o, the letter is rotated clockwise, or counterclockwise, with the center of rotation at point. Finding a Glide eflection Image has vertices (, ), (, ), and (0, 0). Find the image of for a glide reflection where the translation is (, ) (, ) and the reflection line is. First, translate b (, ) (, ). 0 (, ) (, ), or (, ) 0 (,) (, ), or (, ) 0 0 (0, 0) (0,0 ), or (, ) hen, reflect the translated image across the line. he glide reflection image has vertices (, ), (, ), and (, ). Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

147 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Quick heck.. a. eflect the letter across a and then b. escribe the resulting rotation. nswers ma var. he result is a clockwise rotation about point c through an angle of mlacb. a b esson - esson bjectives Identif transformation in tessellations, and figures that will tessellate Identif smmetries in tessellations essellations 00 trand: Geometr opic: Geometr c Vocabular and Ke oncepts. b. Use parallel lines and m. raw between and m. Find the image of for a reflection across line and then across line M. escribe the resulting translation. nswers ma var. he result is a translation twice the distance between and m.. a. Find the image of X under a glide reflection where the translation is (, ) (, ) and the reflection line is. raw the translation first, then the reflection. b. Would the result of part (a) be the same if ou reflected X first, and then translated it? plain. es; rder of steps does not matter in a glide reflection. Geometr esson - X X X m ail otetaking Guide ame lass ate Identifing the ransformation in a essellation Identif the repeating figures and a transformation in the tessellation. repeated combination of an octagon and one adjoining square will completel cover the plane without gaps or overlaps. Use arrows to show a translation. Identifing mmetries in essellations ist the smmetries in the tessellation. tarting at an verte, the tessellation can be mapped onto itself using a 0 rotation, so the tessellation has point smmetr centered at an verte. he tessellation also has translational smmetr, as can be seen b sliding an triangle onto a cop of itself along an of the lines. Quick heck.. plain wh ou can tessellate a plane with an equilateral triangle. he interior angles of an equilateral triangle measure 0. ecause 0 divides evenl into 0, it will tessellate. heorem - ver triangle tessellates. heorem - ver quadrilateral tessellates. tessellation, or tiling, is a repeating pattern of figures that completel covers a plane, without gaps or overlaps. ranslational smmetr is the tpe of smmetr for which there is a translation that maps a figure onto itself. Glide reflectional smmetr is the tpe of smmetr for which there is a glide reflection that maps a figure onto itself. amples. etermining Figures hat Will essellate etermine whether a regular -gon tessellates a plane. plain. ecause the figures in a tessellation do not overlap or leave gaps, the sum of the measures of the angles around an verte must be 0. heck to see whether the measure of an angle of a regular -gon is a factor of 0. ( n ) 0 a Use the formula for the measure of an angle of a regular polgon. n 0 ( ) a ubstitute for n. a implif. ecause is not a factor of 0, a regular -gon will not tessellate a plane. ail otetaking Guide Geometr esson - ame lass ate. Identif a transformation and outline the smallest repeating figure in each tessellation below. a. b. translation nswers ma var. ample: rotation. ist the smmetries in the tessellation. line smmetr, rotational smmetr, glide reflectional smmetr, translational smmetr 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

148 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson 0- esson bjectives Find the area of a parallelogram Find the area of a triangle reas of arallelograms and riangles 00 trand: Measurement opic: Measuring hsical ttributes amples. Finding a Missing imension parallelogram has -in. and -in. sides. he height corresponding to the -in. base is in. Find the height corresponding to the -in. base. First find the area of the parallelogram using the -in. base and its corresponding -in height. Vocabular and Ke oncepts. bh rea of a parallelogram heorem 0-: rea of a ectangle he area of a rectangle is the product of its base and height. bh heorem 0-: rea of a arallelogram he area of a parallelogram is the product of a base and the corresponding height. bh heorem 0-: rea of a riangle he area of a triangle is half the product of a base and the corresponding height. bh base of a parallelogram is an of its sides. he altitude of a parallelogram corresponding to a given base is the segment perpendicular to the line containing that base drawn from the side opposite the base. he height of a parallelogram is the length of its altitude. base of a triangle is an of its sides. he height of a triangle is the length of the altitude to the line containing that base. Geometr esson 0- b h h b h b ltitude ase b h ail otetaking Guide ame lass ate ample. tructural esign he front of a garage is a square ft on each side with a triangular roof above the square. he height of the triangular roof is 0. ft. o the nearest hundred, how much force is eerted b an 0 mi/h wind blowing directl against the front of the garage? Use the formula F 0.00v, where F is the force in pounds, is the area of the surface in square feet, and v is the velocit of the wind in miles per hour. Use the area formulas for rectangles and triangles to find the area of the front of the garage. rea of the square: bh ft rea of the triangle: bh ( )( 0. ). ft he total area of the front of the garage is. 0. ft. Find the force of the wind against the front of the garage. F 0.00v Use the formula for force F. implif. F 00 ound to the nearest hundred. n 0 mi/h wind eerts a force of about 00 lb against the front of the garage. F 0.00( )( ) ubstitute for and for v. Quick heck.. Find the area of the triangle. 0 cm. ritical hinking uppose the bases of the square and triangle in ample are doubled to 0 ft, but the height of each figure stas the same. How is the force of the wind against the building affected? he force is doubled. cm 0. ft ft ft cm cm ( ) ubstitute for b and for h. implif. he area of the parallelogram is in. Use the area to find the height corresponding to the -in. base. bh rea of a parallelogram h ubstitute for and for b. h ivide each side b.. h implif. he height corresponding to the -in base is. in. Finding the rea of a riangle Find the area of XYZ. bh rea of a triangle 0 0 ( )( ) ubstitute for b and for h. implif. XYZ has area cm. Quick heck.. parallelogram has sides cm and cm. he height corresponding to a -cm base is cm. Find the height corresponding to an -cm base.. cm ail otetaking Guide Geometr esson 0- Vocabular and Ke oncepts. cm cm Y X cm heorem 0-: rea of a rapezoid he area of a trapezoid is half the product of the height and the sum of the bases. heorem 0-: rea of a hombus or a Kite he area of a rhombus or a kite is half the product of the lengths of its diagonals. he height of a trapezoid is the perpendicular distance h between the bases. amples. ppling the rea of a rapezoid car window is shaped like the trapezoid shown. Find the area of the window. h(b b ) rea of a cm 0 cm in. cm ame lass ate esson 0- esson bjectives Find the area of a trapezoid Find the area of a rhombus or a kite h(b b ) d d 00 trand: Measurement opic: Measuring hsical ttributes 0 in. Z reas of rapezoids, hombuses, and Kites trapezoid eg ase b h b ase b h b d d eg in. ( )( 0 ) ubstitute for h, 0 for b, and for b. 0 implif. he area of the car window is 0 in.. Geometr esson 0- ail otetaking Guide ail otetaking Guide Geometr esson 0- Geometr ll-in-ne nswers Version

149 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Finding rea Using a ight riangle Find the area of trapezoid. First draw an altitude from verte to that divides trapezoid into a rectangle and a right triangle. he altitude will meet at point X. ecause opposite sides of rectangle X are congruent, X ft and X ft ft ft. the thagorean heorem, X X, so X. aking the square root, X ft. You ma remember that,, is a thagorean triple. h b b implif. he area of trapezoid is ft. ( ) Use the trapezoid area formula. ( )( ) ubstitute for h, for b, and for b. Quick heck.. Find the area of a trapezoid with height cm and bases cm and cm.. cm. In ample, suppose X and change so that m 0 while bases and angles and are unchanged. Find the area of trapezoid..! or. ft Geometr esson 0- Vocabular and Ke oncepts. heorem 0-: rea of a egular olgon he area of a regular polgon is half the product of the apothem and the perimeter. ap ft ft X ft ail otetaking Guide ame lass ate esson 0- esson bjective Find the area of a regular polgon he center of a regular polgon is the center of the circumscribed circle. he radius of a regular polgon is the distance from the center to a verte. he apothem of a regular polgon is the perpendicular distance from the center to a side. amples. Finding ngle Measures his regular heagon has an apothem and radii drawn. Find the measure of each numbered angle. m 0 0 ivide 0 b the number of sides. he apothem bisects the verte angle of m m the isosceles triangle formed b the radii. m ( 0 ) 0 ubstitute 0 for m reas of egular olgons 00 trand: Measurement opic: Measuring hsical ttributes enter adius pothem m 0 ( ) he sum of the measures of the angles of a triangle is 0. a p amples. Finding the rea of a Kite Find the area of kite XYZW. cm Y Find the lengths of the diagonals of kite XYZW. XZ d and YW d X cm d d Use the formula for the area of a kite. ( )( ) ubstitute for d and for d. cm implif. W he area of kite XYZW is cm. Finding the rea of a hombus Find the area of rhombus U. o find the area, ou need to know the lengths of both diagonals. raw diagonal U, and label the intersection of the diagonals point X. X is a right triangle because the diagonals of a rhombus X are perpendicular. U he diagonals of a rhombus bisect each other, so X ft. You can use the thagorean triple,, or the thagorean heorem to conclude that X ft. U 0 ft because the diagonals of a rhombus bisect each other. d d rea of a rhombus ( )( 0 ) ubstitute for d and for d. 0 implif. he area of rhombus U is 0 ft. Quick heck.. Find the area of a kite with diagonals that are in. and in. long. in. ritical hinking In ample, eplain how ou can use a thagorean triple to conclude that XU ft. ail otetaking Guide Geometr esson 0- ft cm Z ame lass ate Finding the rea of a egular olgon librar is in the shape.0 ft of a regular octagon. ach side is.0 ft. he radius of the octagon is. ft. Find the area of the librar to the nearest 0 ft. onsecutive radii form an isosceles triangle, so an apothem bisects the side of the octagon.. ft o appl the area formula ap, ou need to find a and p. tep Find the apothem a..0 ft.0 ft a (.0 ) (. ) thagorean heorem. a olve for a. a a <.. tep Find the perimeter p. p ns Find the perimeter, where n the number of sides of a regular polgon. p ()(.0 ) ubstitute for n and.0 for s, and simplif. tep Find the area. ap rea of a regular polgon.. ( )( ) ubstitute for a and for p.. implif. o the nearest 0 ft, the area is 0 ft. ppling heorem 0- Find the area of an equilateral triangle with apothem cm. eave our answer in simplest radical form. his equilateral triangle shows two radii forming an angle that measures 0 0. ecause the radii and a side form an isosceles triangle, s the apothem bisects the 0 angle, forming two 0 angles. 0 You can use a triangle to find half the length of a side. cm 0 s "? a longer leg "? shorter leg s s "? ubstitute for a. s " Multipl each side b. p ns Find the perimeter. p ( )( " ) " ubstitute for n and! for s, and simplif. ap rea of a regular polgon ( )( " ) ubstitute for a and! for p. " implif. ft m 0, m 0, and m 0 he area of the equilateral triangle is " cm. Geometr esson 0- ail otetaking Guide ail otetaking Guide Geometr esson 0- ll-in-ne nswers Version Geometr

150 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Quick heck.. t the right, a portion of a regular octagon has radii and an apothem drawn. Find the measure of each numbered angle. ml ; ml.; ml. esson 0- esson bjective Find the perimeters and areas of similar figures erimeters and reas of imilar Figures 00 trand: Measurement and umber roperties and perations opics: stems of Measurement; atios and roportional easoning Ke oncepts.. Find the area of a regular pentagon with.-cm sides and an -cm apothem.. he side of a regular heagon is ft. Find the area of the heagon. 0 cm " ft Geometr esson 0- ail otetaking Guide ame lass ate Using imilarit atios enita plants the same crop in two rectangular fields. ach dimension of the larger field is times the dimension of the smaller field. eeding the smaller field costs $. How much mone does seeding the larger field cost? he similarit ratio of the fields is. :, so the ratio of the areas of the fields is (.) : (), or. to. ecause seeding the smaller field costs $, seeding. times as much land costs. $. eeding the larger field costs $. Quick heck.. wo similar polgons have corresponding sides in the ratio :. a. Find the ratio of their perimeters. b. Find the ratio of their areas. : :. he corresponding sides of two similar parallelograms are in the ratio. he area of the larger parallelogram is in.. Find the area of the smaller parallelogram. in.. he similarit ratio of the dimensions of two similar pieces of window glass is :. he smaller piece costs $.0. What should be the cost of the larger piece? $. heorem 0-: erimeters and reas of imilar Figures If the similarit ratio of two similar figures is a b, then a () the ratio of their perimeters is b and a () the ratio of their areas is. b amples. Finding atios in imilar Figures he triangles at the right are similar.find the ratio (larger to smaller) of their perimeters and of their areas. he shortest side of the left-hand triangle has length, and the shortest side of the right-hand triangle has length. From larger to smaller, the similarit ratio is. the erimeters and reas of imilar Figures heorem, the ratio of the perimeters is, and the ratio of the areas is, or. Finding reas Using imilar Figures he ratio of the length of the corresponding sides of two regular octagons is. he area of the larger octagon is 0 ft. Find the area of the smaller octagon. ll regular octagons are similar. ecause the ratio of the lengths of the corresponding sides of the regular octagons is, the ratio of their areas is, or. Write a proportion. 0 0 ross-roduct Use the ropert. ivide each side b. he area of the smaller octagon is ft. ail otetaking Guide Geometr esson 0- ame lass ate ample. Finding imilarit and erimeter atios he areas of two similar pentagons are in. and in.. What is their similarit ratio? What is the ratio of their perimeters? Find the similarit ratio a : b. a he ratio of the areas is a : b. b a b a b implif. ake the square root. he similarit ratio is :. the erimeters and reas of imilar Figures the ratio of the perimeters is also :. heorem, Quick heck.. he areas of two similar rectangles are ft and ft. Find the ratio of their perimeters.! :.. Geometr esson 0- ail otetaking Guide ail otetaking Guide Geometr esson 0- Geometr ll-in-ne nswers Version

151 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson 0- esson bjectives Find the area of a regular polgon using trigonometr Find the area of a triangle using trigonometr Ke oncepts. 00 trand: Measurement opic: Measuring hsical ttributes rigonometr and rea (cos ) 0(sin ) ubstitute for a and p. 0 (cos ) (sin ) implif. 0. Use a calculator. he area is about 0 ft. urveing triangular park has two sides that measure 00 ft and 00 ft and form a angle. Find the area of the park to the nearest hundred square feet. heorem 0-: rea of a riangle Given he area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. rea of amples. Finding rea he radius of a garden in the shape of a regular pentagon is feet. Find the area of the garden. Find the perimeter p and apothem a, and then find the area using the formula ap. 0 ecause a pentagon has five sides, m. and are radii, so. herefore, M M b the H heorem, so m M m. Use the cosine ratio to find a. Use the sine ratio to find M. cos a Use the ratio. sin M a (cos ) olve. M (sin ) Geometr esson 0- bc(sin ) Use M to find p. ecause M M,? M the pentagon is regular, p?.. ecause o p (? M ) 0? M 0? (sin ) 0(sin ). Finall, substitute into the area formula ap. c b ft a M ail otetaking Guide ame lass ate esson 0- esson bjectives Find the measures of central angles and arcs Find circumference and arc length Vocabular and Ke oncepts. 00 trand: Measurement opic: Measuring hsical ttributes ostulate 0-: rc ddition ostulate he measure of the arc formed b two adjacent arcs is the sum of the measures of the two arcs. 0 0 m m m heorem 0-: ircumference of a ircle he circumference of a circle is p times the diameter. pd or pr heorem 0-0: rc ength he length of an arc of a circle is the product of the ratio measure of the arc and the circumference of the circle. 0 length of 0 m 0 0? pr circle is the set of all points equidistant from a given point called the center. center of a circle is the point from which all points are equidistant. ircles and rcs d r r radius is a segment that has one endpoint at the center and the other endpoint on the circle. ongruent circles have congruent radii. diameter is a segment that contains the center of a circle and has both endpoints on the circle. central angle is an angle whose verte is the center of the circle. ircumference of a circle is the distance around the circle. i (π) is the ratio of the circumference of a circle to its diameter. Use heorem 0-: he area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. 00 ft rea side length side length sine of included angle heorem - rea sin ubstitute. rea 0,000 sin ubstitute.. Use a calculator. he area of the park is approimatel,00 ft. Quick heck.. Find the area of a regular octagon with a perimeter of 0 in. Give the area to the nearest tenth.. in.. wo sides of a triangular building plot are 0 ft and ft long. he include an angle of. Find the area of the building plot to the nearest square foot. 0 ft ail otetaking Guide Geometr esson 0- M X 00 ft ame lass ate is a semicircle. is a minor arc. is a major arc. m 0 m m m 0 m semicircle is half of a minor arc is smaller than major arc is greater circle. a semicircle. than a semicircle. djacent arcs are arcs of the same circle that have eactl one point in common. oncentric circles are circles that lie in the same plane and have the same center. rc length is a fraction of a circle s circumference. ongruent arcs are arcs that have the same measure and are in the same circle or in congruent circles. ample. 0 Finding the Measures of rcs Find mxy and mxm in circle mxy mx my rc ddition ostulate 0 0 he measure of a minor arc is the measure mxy m/ X my of its corresponding central angle. 0 mxy 0 ubstitute. Y 0 0 mxy implif. 0 mxm mx mxwm rc ddition ostulate mxm 0 ubstitute. mxm implif. Quick heck.. Use the diagram in ample. Find m Y, mymw, mmw 0, and mxmy. 0; 0; ; W Geometr esson 0- ail otetaking Guide ail otetaking Guide Geometr esson 0- ll-in-ne nswers Version Geometr

152 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. esson 0- reas of ircles and ectors ppling ircumference circular swimming pool feet in diameter will be enclosed in a circular fence ft from the pool. What length of fencing material is needed? ound our answer to the net whole number. he pool and the fence are concentric circles. he diameter of the pool is ft, so the diameter of the fence is ft. Use the formula for the circumference of a circle to find the length of fencing material needed. pd Formula for the circumference of a circle π ubstitute. <.( ) Use. to approimate π. <. implif. bout ft of fencing material is needed. Finding rc ength Find the length of in circle M in terms of π. 0 ecause 0, m rc ddition ostulate length of m 0? pr rc ength ostulate 0 0 π( ) ubstitute. π he length of is π cm. Quick heck.. he diameter of a biccle wheel is in. o the nearest whole number, how man revolutions does the wheel make when the biccle travels 00 ft? revolutions. Find the length of a semicircle with radius. m in terms of π..p m Geometr esson 0- ft ft ft 0 cm M ail otetaking Guide ame lass ate Finding the rea of a ector of a ircle Find the area of sector. eave our answer in terms of π. m area of sector πr 0 π 0 π he area of sector is 0π m π ( ) Quick heck.. How much more pizza is in a -in.-diameter pizza than in a -in. pizza? about in.. ritical hinking circle has a diameter of 0 cm. What is the area of a sector bounded b a 0 major arc? ound our answer to the nearest tenth.. cm m 00 esson bjective Find the areas of circles, sectors, and segments of circles Vocabular and Ke oncepts. heorem 0-: rea of a ircle he area of a circle is the product of p and the square of the radius. pr heorem 0-: rea of a ector of a ircle measure of the arc he area of a sector of a circle is the product of the ratio 0 and the area of the circle.. 0 rea of sector m 0? pr sector of a circle is a region bounded b two radii and their intercepted arc. segment of a circle is the part bounded b an arc and the segment joining its endpoints. amples. ppling the rea of a ircle circular archer target has a -ft diameter. It is ellow ecept for a red bull s-ee at the center with a -in. diameter. Find the area of the ellow region to the nearest whole number. First find the areas of the archer target and the red bull s-ee. he radius of the archer target is ( ft) ft in. he area of the archer target is πr π( ) p in. he radius of the red bull s-ee region is ( in.) in. he area of the red region is πr π( ) p in. area of area of area of archer target red region ellow region π π π.0 he area of the ellow region is about in. 00 trand: Measurement opic: Measuring hsical ttributes implif. r ail otetaking Guide Geometr esson 0- r egment of a circle ector of a circle ame lass ate ample. Finding the rea of a egment of a ircle Find the area of the shaded segment. ound our answer to the nearest tenth. tep Find the area of sector. 0 ft 0 m area of sector? πr Use the formula for area of a sector. 0 0? 0 π ( ) r ubstitute.? π π implif. tep Find the area of. You can use a triangle to find the height h of and the length of. h hpotenuse? shorter leg h ivide each side b.!? " longer leg "? shorter leg " Multipl each side b. has base " ft " ft, or " ft, and height ft. bh rea of a triangle. ( " )( ) ubstitute " for b and for h. " implif. tep ubtract the area of from the area of sector to find the area of the segment of the circle. area of segment π " area of segment<.0 Use a calculator. o the nearest tenth, the area of the shaded segment is. ft. Quick heck.. circle has a radius of cm. Find the area of the smaller segment of the circle determined b a 0 arc. ound our answer to the nearest tenth. ft ft 0 0 ft ft 0.0 cm 00 Geometr esson 0- ail otetaking Guide ail otetaking Guide Geometr esson Geometr ll-in-ne nswers Version

153 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson 0- esson bjective Use segment and area models to find the probabilities of events Geometric robabilit 00 trand: ata nalsis and robabilit opic: robabilit ample. Finding robabilit Using rea circle is inscribed in a square target with 0-cm sides. Find the probabilit that a dart landing randoml within the square does not land within the circle. Find the area of the square. 0 cm Vocabular. Geometric probabilit is a model in which ou let points represent outcomes. s 0 00 cm Find the area of the circle. ecause the square has sides of length 0 cm, the circle s diameter is 0 cm, so its radius is 0 cm. πr π( 0 ) 00π cm ample. Finding robabilit Using egments gnat lands at random on the edge of the ruler below. Find the probabilit that the gnat lands on a point between and 0. he length of the segment between and 0 is 0. he length of the ruler is. (landing between and 0) length of favorable segment length of entire segment Quick heck.. point on is selected at random. What is the probabilit that it is a point on? Geometr esson 0- ail otetaking Guide ame lass ate ample. ppling Geometric robabilit o win a prize, ou must toss a quarter so that it lands entirel within the outer region of the circle at right. Find the probabilit that this happens with a quarter of radius in. ssume that the in. quarter is equall likel to land anwhere completel inside the large circle. he center of a quarter with a radius of in. must land at least in. in. beond the boundar of the inner circle in order to lie entirel outside the inner circle. ecause the inner circle has a radius of in., the quarter must land outside the circle whose radius is in. in., or in. Find the area of the circle with a radius of in. pr p <. in. imilarl, the center of a quarter with a radius of in. must land at least in. within the outer circle. ecause the outer circle has a radius of in., the quarter must land inside the circle whose radius is in. in., or in. Find the area of the circle with a radius of in. pr p <. in. Use the area of the outer region to find the probabilit that the quarter lands entirel within the outer region of the circle... (outer region) area of outer circle area of large circle..00 (outer region) 0.. he probabilit that the quarter lands entirel within the outer region of the circle is about 0. or. %. Quick heck.. ritical hinking Use ample. uppose ou toss 00 quarters. Would ou epect to win a prize? plain. Yes; theoreticall ou should win. times out of 00. Find the area of the region between the square and the circle. (00 00π ) cm Use areas to calculate the probabilit that a dart landing randoml in the square does not land within the circle. Use a calculator. ound to the nearest thousandth. area between square and circle (between square and circle) area of square 00 00π 00 π he probabilit that a dart landing randoml in the square does not land within the circle is about. %. Quick heck.. Use the diagram in ample. If ou change the radius of the circle as indicated, what then is the probabilit of hitting outside the circle? a. ivide the radius b. about 0.% b. ivide the radius b. about.% 0. ail otetaking Guide Geometr esson 0- ame lass ate esson - esson bjective ecognize polhedra and their parts Visualize cross sections of space figures Vocabular and Ke oncepts. uler s Formula he numbers of faces (F), vertices (V), and edges () of a polhedron are related b the formula F V. polhedron is a three-dimensional figure whose surfaces are polgons. face is a flat surface of a polhedron in the shape of a polgon. n edge is a segment that is formed b the intersection of two faces. verte is a point where three or more edges intersect. cross section is the intersection of a solid and plane. amples 00 trand: Geometr opic: imension and hape 0 pace Figures and ross ections ectangular Faces dge Verte prism 0 Geometr esson 0- ail otetaking Guide ail otetaking Guide Geometr esson - 0 ll-in-ne nswers Version Geometr

154 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. Identifing Vertices, dges, and Faces How man vertices, edges, and faces are there in the polhedron shown? Give a list of each. ample. Verif uler s Formula for the two-dimensional net of the solid in ample. here are four vertices:,,, and. ount the regions: F here are si edges:,,,,, and. ount the vertices: V 0 here are four faces: k, k, k, and k. Using uler s Formula Use uler s Formula to find the number of edges on a solid with faces and vertices. F V uler s Formula ubstitute the number of faces and vertices. implif. solid with faces and vertices has Quick heck. Geometr esson - edges.. ist the vertices, edges, and faces of the polhedron.. Use uler s Formula to find the number of edges on a polhedron with eight triangular faces.,,, U, V;, U,, V, VU, V, U, U, ; ku, ku, k, kvu, kvu, kv edges U V ail otetaking Guide ame lass ate esson - esson bjectives Find the surface area of a prism Find the surface area of a clinder Vocabular and Ke oncepts. 00 trand: Measurement opic: Measuring hsical ttributes heorem -: ateral and urface rea of a rism he lateral area of a right prism is the product of the perimeter of the base and the height... ph he surface area of a right prism is the sum of the lateral area the areas of the two bases..... erimeter of base c c a b c d d d ase a b c d a b h a b c h urface reas of risms and linders d h d ase c ateral rea erimeter Height ateral rea ph urface rea.. rea of a base heorem -: ateral and urface rea of a linder he lateral area of a right clinder is the product of the circumference of the base and the height of the clinder... πrh, or.. πdh he surface area of a right clinder is the sum of the lateral area and the areas of the two circular bases....., or.. πrh πr h πr ateral rea r h urface rea r r and rea of a base π r ount the segments: Quick heck.. he figure at the right is a trapezoidal prism. a. verif uler s Formula F V for the prism. b. raw a net for the prism. ossible answer: c. Verif uler s Formula F V for our two-dimensional net. ail otetaking Guide Geometr esson - ame lass ate prism is a polhedron with eactl two congruent, parallel faces. he bases of a polhedron are are the parallel faces. ateral faces are the faces on a prism that are not bases. n altitude is a perpendicular segment that joins the planes of the bases. he height is the length of the altitude. he lateral area of a prism is the sum of the areas of the lateral faces. he surface area of a prism or a clinder is the sum of the lateral area and the areas of the two bases. right prism is a prism whose lateral faces are rectangular regions and a lateral edge is an altitude. n oblique prism is a prism whose lateral faces are not perpendicular to the bases. ases right right h h h h prisms clinder is a three-dimensional figure with two congruent circular bases that lie in parallel planes. In a clinder, the bases are parallel circles. clinders he lateral area of a clinder is the area of the curved surface. right clinder is one where the segment joining the centers of the bases is an altitude. n oblique clinder is one in which the segment joining the centers of the bases is not perpendicular to the bases. ases ateral faces entagonal riangular oblique oblique h h 0 prism ateral edges ases prism prism clinder 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 Geometr ll-in-ne nswers Version

155 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. Using Formulas to Find urface rea Find the surface area of a 0-cmhigh right prism with triangular bases having -cm edges. ound to the nearest whole number. Use the formula.. ph to find the lateral area and the formula.... to find the surface area of the prism. he area of the base is ap, where a is the apothem and p is the perimeter. he triangle has sides of length cm, so p? cm, or cm. 0 a cm cm 0 a 0 cm cm? "!? shorter leg longer leg a ivide both sides b ". " Geometr esson - Use the diagram of the base. Use triangles to find the apothem. "! a? " ationalize the denominator. " " ap? "? ubstitute for a and for p.! " implif. he area of each base of the prism is cm.!.... Use the formula for surface area. ph ph ubstitute for.. 0! 0! 0 implif.! < 0. Use a calculator. ounded to the nearest whole number, the surface area is cm. ( )( ) ( ) ubstitute for p, for h and for. Vocabular and Ke oncepts. heorem -: ateral and urface rea of a egular ramid he lateral area of a regular pramid is half the product of the perimeter of the base and the slant height... p he surface area of a regular pramid is the sum of the lateral area and the area of the base..... heorem -: ateral and urface rea of a one he lateral area of a right cone is half the product of the circumference of the base and the slant height...? πr?, or.. pr he surface area of a right cone is the sum of the lateral area and the area of the base..... ail otetaking Guide ame lass ate esson - esson bjectives Find the surface area of a pramid Find the surface area of a cone 00 trand: Measurement opic: Measuring hsical ttributes urface reas of ramids and ones regular pramid is a pramid whose base is a regular polgon and whose lateral faces are congruent isosceles triangles. he altitude of a pramid or a cone is the perpendicular segment from the verte to the plane of the base. he height of a pramid or a cone is the length of ateral Verte the altitude. edge ateral face he slant height of a regular pramid is the length ltitude of the altitude of a lateral face. ase ase edge he lateral area of a pramid is is the sum of the egular ramid areas of the congruent lateral faces. he surface area of a pramid is is the sum of the lateral area and the area of the base. r Finding urface rea of linders compan sells cornmeal and barle in clindrical containers. he diameter of the base of the -in.-high cornmeal container is in. he diameter of the base of the -in.-high barle container is in. Which container has the greater surface area? Find the surface area of each container. emember that r d. ornmeal ontainer arle ontainer.... Use the formula for surface area..... ubstitute the formulas for lateral prh p r p r h pr area of a clinder and area of a circle. π( )( ) π( ) ubstitute for r and h. π( )( ) π( ) p p implif. p p p ombine like terms. p ecause π in. π in., the barle container has the greater surface area. Quick heck.. Use formulas to find the lateral area and surface area of the prism. m ; about m. Find the surface area of a clinder with height 0 cm and radius 0 cm in terms of π. 00p cm. he compan in ample wants to make a label to cover the cornmeal container. he label will cover the container all the wa around, but will not cover an part of the top or bottom. What is the area of the label to the nearest tenth of a square inch?. in. m m ail otetaking Guide Geometr esson - ame lass ate right cone is a cone in which the altitude is perpendicular to the plane of the base. he slant height of a cone is the distance from the verte to the edge of the base. he lateral area of a cone is the area of the curved lateral surface. he surface area of a cone the sum of the lateral area and the area of the base. amples. Finding urface rea of a ramid Find the surface area of a square pramid with base edges. ft and slant height ft. he perimeter p of the square base is. ft, or 0 ft. You are given ft and ou found that p 0 ft, so ou can find the lateral area ft ft ( 0 )( ) ubstitute. 0 p Use the formula for lateral area of a pramid. implif. Find the area of the square base. ecause the base is a square with side length. ft, s Use the formula for surface area of a pramid ubstitute. implif. / lant height h r ight ltitude ase cone he surface area of the square pramid is. ft. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

156 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Finding ateral rea of a one eandre uses paper cones to cover her plants in the earl spring. he diameter of each cone is ft, and its height is. ft. How much paper is in the cone? ound our answer to the nearest tenth. Use the formula.. pr to find the lateral area of the cone. esson - esson bjectives Find the volume of a prism Find the volume of a clinder Volumes of risms and linders 00 trand: Measurement opic: Measuring hsical ttributes he cone s diameter is ft, so its radius r is 0. ft. he altitude of the cone, radius of the base, and slant height form a right triangle. Use the thagorean heorem to find the slant height ". Find the lateral area... p r Use the formula for lateral area of a cone... p 0. ". 0. "... <. Use a calculator. he lateral area of the cone is about. ft. ( ) ubstitute for r and for. Quick heck.. Find the surface area of a square pramid with base edges m and slant height m. m. Find the lateral area of a cone with radius in. and height 0 in. to the nearest square inch. in. Geometr esson - ail otetaking Guide ame lass ate Finding Volume of a riangular rism Find the volume of the prism. he prism is a right triangular prism with triangular bases. he base of the triangular prism is a right triangle where one leg is the base and the other leg is the altitude. Use the thagorean heorem to calculate the length of the other leg. area of the base to find the volume of the prism. V h Use the formula for the volume of a prism. V 0? 0 ubstitute. V 00 implif. he volume of the triangular prism is 00 m. he area of the base is bh ( 0 )( ) 0. Use the Finding Volume of a omposite Figure Find the volume of the composite space figure. You can use three rectangular prisms to find the volume. cm cm cm cm cm cm cm cm cm cm ach prism s volume can be found using the formula V h. Volume of rism I h (? )? Volume of rism II h (? )? Volume of rism III h (? )? um of the volumes cm 00 0 m cm cm cm m 0 m cm cm cm 0 cm Vocabular and Ke oncepts. heorem -: avalieri s rinciple If two space figures have the same height and the same cross sectional area at ever level, then the have the same volume. heorem -: Volume of a rism he volume of a prism is the product of the area of a base and the height of the prism. V h heorem -: Volume of a linder he volume of a clinder is the product of the and the height of the clinder. V h, or V pr h area of the base Volume is the space that a figure occupies. composite space figure is a three-dimensional figure that is the combination of two or more simpler figures. amples. Finding Volume of a linder Find the volume of the clinder. eave our answer in terms of π. he formula for the volume of a clinder is V pr h. he diagram shows h and d, but ou must find r. r d V p p? r h? p Use the formula for the volume of a clinder. ubstitute. implif. he volume of the clinder is π ft. ail otetaking Guide Geometr esson - ame lass ate Quick heck.. Find the volume of the triangular prism. 0 m. he clinder shown is oblique. a. Find its volume in terms of π. p m b. Find its volume to the nearest tenth of a cubic meter. 0. m. Find the volume of the composite space figure. in. h ft 0 m m in. r m m in. in. h ft m in. in. he volume of the composite space figure is 00 cm. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

157 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Find the volume of a pramid Find the volume of the cone Ke oncepts. Volumes of ramids and ones 00 trand: Measurement opic: Measuring hsical ttributes Finding Volume of a ramid Find the volume of a square pramid with base edges m and slant height m. he altitude of a right square pramid intersects the base at the center of the square. ecause each side of the square base is m, the leg of the right triangle along the base is m, as shown. h m m heorem -: Volume of a ramid he volume of a pramid is one third the product of the area of the base and the height of the pramid. h V h heorem -: Volume of a one he volume of a cone is one third the product of the area of the base and the height of the cone. V h h, or V pr h r amples. Finding Volume of a ramid Find the volume of a square pramid with base edges cm and height cm. ecause the base is a square,?. V h Use the formula for volume of a pramid. ( )( ) ubstitute for and for h. V 0 implif. he volume of the square pramid is 0 cm. Geometr esson - ail otetaking Guide ame lass ate Quick heck.. Find the volume of a square pramid with base edges m and slant height m. 0 m. Find the volume of each cone in terms of π and also rounded as indicated. a. to the nearest cubic meter m m p m ; m b. to the nearest cubic millimeter mm mm p mm ;, mm. small child s teepee is ft tall and ft in diameter. Find the volume of the teepee to the nearest cubic foot. ft h m m tep Find the height of the pramid. h Use the thagorean heorem. h implif. h ubtract from each side. h Find the square root of each side. tep Find the volume of the pramid. Use the formula for V h the volume of a pramid. (? ) ubstitute. 0 implif. he volume of the square pramid is 0 m. Finding Volume of an blique one Find the volume of the oblique cone in terms of π. r d in. V p r h Use the formula for the volume of a cone. ( p )( ) ubstitute for r and for h. p implif. he volume of the cone is p in.. m ail otetaking Guide Geometr esson - in. in. ame lass ate esson - esson bjective Find the surface area and volume of a sphere Vocabular and Ke oncepts. heorem -0: urface rea of a phere he surface area of a sphere is four times the product of p and the square of the radius of the sphere... pr heorem -: Volume of a phere he volume of a sphere is four thirds the product of p and the cube of the radius of the sphere... pr sphere is the set of all points in space equidistant from a given point. he center of a sphere is the given point from which all points on the sphere are equidistant. he radius of a sphere is a segment that has one endpoint at the center and the other endpoint on the sphere. he diameter of a sphere is a segment passing through the center with endpoints on the sphere. great circle is the intersection of a plane and a sphere containing the center of the sphere. great circle divides a sphere into two congruent hemispheres. he circumference of a sphere is the circumference of its 00 trand: Measurement opic: Measuring hsical ttributes urface reas and Volumes of pheres center d circumference Hemispheres r sphere r r diameter Great circle great circle. 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

158 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. ample. Finding urface rea he circumference of a rubber ball is cm. alculate its surface area to the nearest whole number. Using Volume to Find urface rea he volume of a sphere is in.. Find its surface area to the nearest tenth. tep First, find the radius. p r Use the formula for circumference. pr ubstitute for. r olve for r. p tep Use the radius to find the surface area... p r Use the formula for surface area of a sphere. p? ( ) ubstitute for r. p p Geometr esson - implif. Use a calculator. o the nearest whole number, the surface area of the rubber ball is cm. Finding Volume Find the volume of the sphere. eave our answer in terms of π. V p Use the formula for volume of a sphere. 0? ubstitute r. implif. he volume of the sphere is 00p cm. Quick heck. p < r 00 p p.. Find the surface area of a sphere with d in. Give our answer in two was, in terms of π and rounded to the nearest square inch. p in., in. Vocabular and Ke oncepts. heorem -: reas and Volumes of imilar olids If the similarit ratio of two similar solids is a : b, then () the ratio of their corresponding areas is a : b, and () the ratio of their volumes is a : b. 0 cm ail otetaking Guide ame lass ate esson - esson bjective Find relationships between the ratios of the areas and volumes of similar solids imilar solids have the same shape and all of their corresponding parts are proportional. he similarit ratio of two similar solids is the ratio of their corresponding linear dimensions. amples. Identifing imilar olids re the two solids similar? If so, give the similarit ratio. in. in. in. in. 00 trand: Measurement opic: stems of Measurement oth figures have the same shape. heck that the ratios of the corresponding dimensions are equal. he ratio of the radii is, and the ratio of the heights is. he cones are not similar because. reas and Volumes of imilar olids p p πr Use the formula for volume of a sphere. ubstitute. olve for r. Find the of each side.... Use r,.. πr, and a calculator. o the nearest tenth, the surface area of the sphere is. in.. Quick heck. V r r p r cube root. Find the surface area of a spherical melon with circumference in. ound our answer to the nearest ten square inches. 00 in.. Find the volume to the nearest cubic inch of a sphere with diameter 0 in.,0 in.. he volume of a sphere is 00 ft. Find the surface area to the nearest tenth.. ft ail otetaking Guide Geometr esson - ame lass ate Finding the imilarit atio Find the similarit ratio of two similar clinders with surface areas of π ft and π ft. Use the ratio of the surface areas to find the similarit ratios. a p he ratio of the surface areas is :. b a b p a b a b implif. ake the he similarit ratio is :. a b implif. of each side. Using a imilarit atio wo similar square pramids have volumes of cm and cm. he surface area of the larger pramid is cm. Find the surface area of the smaller pramid. tep Find the similarit ratio. a he ratio of the volumes is :. b a b a ake the of each side. b cube root tep Use the similarit ratio to find the surface area of the smaller pramid. he ratio of the surface areas is a : b. 0 implif. ubstitute olve for. implif. he surface area of the smaller pramid is 0 cm.? square root the larger pramid. for, the surface area of Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

159 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Quick heck.. re the two clinders similar? If so, give the similarit ratio. es; : 0 esson - esson bjectives Use the relationship between a radius and a tangent Use the relationship between two tangents from one point angent ines 00 trand: Geometr opic: elationships mong Geometric Figures Vocabular and Ke oncepts.. Find the similarit ratio of two similar prisms with surface areas m and m.. he volumes of two similar solids are m and 0 m. he surface area of the larger solid is 0 m. What is the surface area of the smaller solid? : 0 m Geometr esson - ail otetaking Guide ame lass ate amples. Finding ngle Measures is tangent to at point. Find the value of. ecause is tangent to, must be a right angle. Use the riangle ngle-um heorem to find. m m m 0 riangle ngle-um heorem 0 0 ubstitute. 0 implif. olve. ppling angent ines belt fits tightl around two circular pulles, as shown. Find the distance between the centers of the pulles. ound our X answer to the nearest tenth. raw. hen draw parallel to ZW to form rectangle WZ. ecause Z is a radius of, Z cm. Z ecause opposite sides of a rectangle have the same measure, W cm and cm. ecause is the supplement of a right angle, is also a right angle, and is a right triangle. ecause the radius of is cm, cm. thagorean heorem ubstitute. implif.. Use a calculator to find the square root. he distance between the centers of the pulles is about. cm. Quick heck.. is tangent to. Find the value of. cm cm cm W Y cm heorem - If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangenc. * ) ' heorem - If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. is tangent to. * ) heorem - he two segments tangent to a circle from a point outside the circle are congruent. > tangent to a circle is a line, segment, or ra in the plane of the circle that intersects the circle in eactl one point. he point of tangenc is the point where a circle and a tangent intersect. tangent point of tangenc triangle is inscribed in a circle if all vertices of the triangle lie on the circle. triangle is circumscribed about a circle if each side of the triangle is tangent to the circle. ail otetaking Guide Geometr esson - amples. Finding a angent has radius. oint is outside such that, and point is on such that. Is tangent to at? plain. For to be tangent to at, must be a right angle, must be a right triangle, and. Is a right triangle? ubstitute. implif. ecause, is not tangent to at. ircles Inscribed in olgons is inscribed in quadrilateral XYZW. Find the perimeter of XYZW. Y ft ft Z ft X U ft W UX X ft Y Y ft heorem -, two segments tangent to a circle from a point outside the circle are Z Z ft congruent. WU W ft YZ WX p XY ZW efinition of perimeter p Y Z W UX X Y Z WU egment ddition ostulate ubstitute. implif. he perimeter is ft. ame lass ate Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

160 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate Quick heck.. belt fits tightl around two circular pulles, as shown. Find the distance between the centers of the pulles. in. in. in. esson - esson bjectives Use congruent chords, arcs, and central angles ecognize properties of lines through the center of a circle hords and rcs 00 trand: Geometr opic: elationships mong Geometric Figures Vocabular and Ke oncepts.. In ample, if,, and, is tangent to at? plain. o;. is inscribed in Q. Q has a perimeter of cm. Find QY. cm 0 about. in. Geometr esson - cm X Q Y Z cm ail otetaking Guide ame lass ate amples. Using heorem - In the diagram, radius X bisects. What can ou conclude? X lx b the definition of an angle bisector. X X because congruent central angles have congruent chords. 0 0 X X because congruent chords have congruent arcs. Using heorem - Find. Q Q egment ddition ostulate Q ubstitute. Q implif. Q hords that are equidistant from the center of a circle are congruent. ubstitute for Q. Using iameters and hords and Q are points on. he distance from to Q is in., and Q in. Find the radius of. he distance from the center of to Q is measured along a perpendicular line. diameter that is perpendicular M Q to a chord bisects the chord. M ( ) ubstitute. M M Use the thagorean heorem. r ubstitute. r implif. r Find the square root of each side. Q in. r M Q in. X heorem - Within a circle or in congruent circles () ongruent central angles have congruent chords. () ongruent chords have congruent arcs. () ongruent arcs have congruent central angles. heorem - Within a circle or in congruent circles () hords equidistant from the center are congruent. () ongruent chords are equidistant from the center. heorem - In a circle, a diameter that is perpendicular to a chord bisects the chord and its arcs. heorem - In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord. heorem - In a circle, the perpendicular bisector of a chord contains the of the circle. chord is a segment whose endpoints are on a circle. center ail otetaking Guide Geometr esson - ame lass ate Quick heck.. If ou are given that F in the circles, what can ou conclude? 0 0 l l; F. Find the value of in the circle at the right.. Use the circle at the right. a. Find the length of the chord to the nearest unit. about b. Find the distance from the midpoint of the chord to the midpoint of its minor arc... Q F he radius of is in. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

161 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Find the measure of an inscribed angle Find the measure of an angle formed b a tangent and a chord Vocabular and Ke oncepts. heorem -: Inscribed ngle heorem he measure of an inscribed angle is half the measure of its intercepted arc. Geometr esson - m heorem -0 he measure of an angle formed b a tangent and a chord is half the measure of the intercepted arc. m m orollaries to the Inscribed ngle heorem. wo inscribed angles that intercept the same arc are congruent.. n angle inscribed in a semicircle is a right angle.. he opposite angles of a quadrilateral inscribed in a circle are supplementar. n inscribed angle has a verte that is on a circle and sides that are chords of the circle. n intercepted arc is an arc with endpoints on the sides of an inscribed angle and its other points in the interior of the angle. 00 trand: Geometr opic: elationships mong Geometric Figures Inscribed ngles m Intercepted arc Inscribed angle ail otetaking Guide ame lass ate * ) U Using heorem -0 and are diameters of circle. is tangent to at point. Find m and m. 0 m m heorem m m U m U rc ddition ostulate ubstitute 0 for mu m ( 0 ) and for mu. m implif. Use the properties of tangents to find m. tangent is perpendicular to m 0 the radius of a circle at its point of tangenc. m m m ngle ddition ostulate 0 m ubstitute. m implif. Quick heck.. In ample, find m F. 0. For the diagram at the right, find the measure of each numbered angle. ml 0, ml 0, ml 0, ml 0. In ample, describe two was to find m using heorem -0. ml mlu mlu mu 0 m 0 m U (0) 0 U 0 0 amples. Using the Inscribed ngle heorem Find the values of and. mf Inscribed ngle heorem 0 0 (m m F ) rc ddition ostulate 0 0 ( ) ubstitute. implif. 0 ecause FG is the intercepted arc of, ou need to find mfg in order to find mfg. he arc measure of a circle is 0, so 0 mfg mfg Inscribed ngle heorem 0 0 (m F m FG ) rc ddition ostulate 0 0 ( ) ubstitute. implif. Using orollaries to Find ngle Measures Find the values of a and b. orollar to the Inscribed ngle heorem, an angle inscribed in a semicircle is a right angle, so a 0. b he sum of the measures of the three angles of the triangle inscribed in is 0. herefore, the angle whose intercepted arc has measure b must have measure 0 0 or. ecause the inscribed angle has half the measure of the intercepted arc, the intercepted arc has twice the measure of the inscribed angle, so b ( ). ail otetaking Guide Geometr esson - ame lass ate esson - esson bjectives Find the measures of angles formed b chords, secants, and tangents Find the lengths of segments associated with circles Vocabular and Ke oncepts. heorem - he measure of an angle formed b two lines that () intersect inside a circle is half the sum of the measures of the intercepted arcs. m () intersect outside a circle is half the heorem - For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along an line through the point and circle. I. a c b d a b c d ( ) difference of the measures of the intercepted arcs. m ( ) secant is a line that intersects a circle at two points. II. 00 trand: Geometr z w (w )w ( z) III. 0 0 opic: elationships mong Geometric Figures z a ( z) t t G 0 F ngle Measures and egment engths secant Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

162 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate amples. Finding rc Measures n advertising agenc wants a frontal photo of a fling saucer ride at an amusement park. he photographer stands at the verte of the angle formed b tangents to the fling saucer. What is the measure of the arc that will ) be in the ) photograph? In the 0 diagram, the photographer stands at point. X and Y intercept minor arc XY and major arc XY. 0 et mxy. hen mxy 0. 0 m ( mxy mxy ) heorem -() 0 [( ) ] ubstitute. ( 0 ) implif. 0 istributive ropert. 0 dd to both sides. 0 olve for. 0 Geometr esson - arc will be in the advertising agenc s photo. Finding egment engths tram travels from point to point along the arc of a circle with a radius of ft. Find the shortest distance from point to point. he perpendicular bisector of the chord contains the center of the circle. ecause the radius is ft, the diameter is 0 ft. he length of the other segment along the diameter is 0 ft 0 ft, or 00 ft heorem -() 0,000 Multipl. 00 olve for. he shortest distance from point to point is or 00 ft. 00 ft 0 ft X Y ail otetaking Guide ame lass ate esson - esson bjectives Write an equation of a circle Find the center and radius of a circle Ke oncepts. 00 trand: Geometr opic: osition and irection heorem - he standard form of an equation of a circle with center and radius r is ( h) ( k) r. amples. Writing the quation of a ircle Write the standard equation of a circle with center (, 0) and radius ". h k r ( ) ( ) tandard form " 0, 0 " for r. ( ) implif. [ ( )] ( ) ( ) ubstitute ( ) for (h, k) and ircles in the oordinate lane Using the enter and a oint on a ircle Write the standard equation of a circle with center (, ) that passes through the point (, ). First find the radius. r #( h ) ( k ) Use the istance Formula to find r. #( ) ( ) ubstitute (, ) for (h, k) and (, ) for (, ). #( 0 ) ( ) implif. # 00 # hen find the standard equation of the circle with center (, ) and radius. (h, k) (, ) r (h, k) Quick heck.. Find the value of w. 0. ritical hinking o photograph a 0 arc of the fling saucer in ample, should ou move toward or awa from the object? What angle should the tangents form? awa; 0. Find the value of to the nearest tenth... he basis of a design of a rotor for a Wankel engine is an equilateral triangle. ach side of the triangle is a chord to an arc of a circle. he opposite verte of the triangle is the center of the arc. In the diagram at the right, each side of the equilateral triangle is in. long. a. Use what ou know about equilateral triangles and find the value of. ( ") in. b. ritical hinking omplete the circle with the given center. hen use heorem - to find the value of. how that our answers to parts (a) and (b) are equal. in. ; ( ").0 " " in. enter ail otetaking Guide Geometr esson - w ame lass ate Graphing a ircle Given Its quation Find the center and radius of the circle with equation ( ) ( ). hen graph the circle. ( ) ( ) ( ) ( ) elate the equation to the ( ) ( h) ( k) r. standard form c c c h k r he center is (, ) and the radius is. ppling the quation of a ircle diagram locates a radio tower at (, ) on a coordinate grid where each unit represents mi. he radio signal s range is 0 mi. Find an equation that describes the position and range of the tower. he center of a circular range is at (, ), and the radius is 0. ( h) ( k) r Use standard form. ( ) [ ( )] 0 ubstitute. ( ) ( ) his is an equation for position 00 and range of the tower. Quick heck.. Write the standard equation of each circle. a. center (, ); radius b. center (, ); radius " ( ) ( ) ( ) ( ) rc ( h ) ( k ) r tandard form ( ) ( ) ( ) ubstitute (, ) for (h, k) and for r. ( ) ( ) implif. 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 Geometr ll-in-ne nswers Version

163 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate. Write the standard equation of the circle with center (, ) that passes through the point (, ). esson - ocus: et of oints ( ) ( ) esson bjective raw and describe a locus 00 trand: Geometr opic: imension and hape. Find the center and radius of the circle with equation ( ) ( ) 00. hen graph the circle. center: (, ); radius: 0. When ou make a call on a cellular phone, a tower receives the call. In the diagram, the centers of circles,, and are locations of cellular telephone towers. Write an equation that describes the position and range of ower. Geometr esson (, ) ail otetaking Guide ame lass ate rawing a ocus for wo onditions oint is 0 cm from point Q. escribe the locus of points in a plane that are cm from point and cm from point Q. cm cm 0 cm Q cm cm he set of points in a plane that are cm from point is a circle with center and radius cm. he set of points in a plane that are cm from point Q is a circle with center Q and radius cm. ecause the sum ( cm) of the radii is greater than 0 cm, the circles overlap b cm on Q. he circles intersect at two points, so the locus is the two points where the circles intersect. escribing a ocus in pace escribe the locus of points in space that are cm from a plane M. cm cm M Imagine plane M as a horizontal plane. ecause distance is measured along a perpendicular segment from a point to a plane, the locus of points in space that are cm from a plane M are a plane cm above and parallel to plane M and another plane cm below and parallel to plane M. Vocabular. locus is a set of points, all of which meet a stated condition. amples. escribing a ocus in a lane raw and describe the locus of points in a plane that are. cm from with radius. cm. Use a compass to draw with radius. cm.. cm. cm he locus of points in the interior of that are. cm from is the center. he locus of points eterior to that are. cm from is a circle with center and radius cm. he locus of points in a plane that are. cm from a point on with radius. cm is point and a circle with radius cm and center. ail otetaking Guide Geometr esson - ame lass ate Quick heck. * ). raw and describe the locus: In a plane, the points cm from line XY. * ) * ) wo lines parallel to XY, each cm from XY. cm. raw the locus: In a plane, the points equidistant from two points X and Y and cm from the midpoint of XY. } cm X Y cm. raw and describe each locus. a. In a plane, the points that are equidistant from two parallel lines. line midwa between the two parallel lines, parallel to and equidistant from each. b. In space, the points that are equidistant from two parallel planes. plane midwa between the two parallel planes, parallel to and equidistant from each. } X cm Y Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

164 Geometr: ll-in-ne nswers Version (continued) hapter ractice -.,.,. -,. ample: or. or. Y or. an heagon. a. angle. 0. ample: he farther out ou go, the closer the ratio gets to a number that is approimatel ,,,,,,, Guided roblem olving -. he pattern is easier to visualize.. he graph will go up.. Use Years for the horizontal ais.. Use umber of tations for the vertical ais.. increasing. greater. Yes. ince the number of stations increases steadil from 0 to 000, we can be confident that the number of stations in 00 will be greater than in atterns are necessar to reach a conclusion through inductive reasoning.. (an list of numbers without a pattern would appl),;,;,;,; 0, ractice Front Front Front ight Front Front ight ight ight Front op op.,,.... ight ight Guided roblem olving -. he represent three-dimensional objects on a twodimensional surface.. nine. ee the figure in, below... Yes. It is similar to the foundation drawing, ecept there are no numbers.. no. op. Yes.. ractice -. * ). an two of the following:,,,,,,,. es. no. es. es * ) * ). es. es. G 0. M. the empt set.. ample: plane. * ) K. no. es. the empt set. no. es 0. es Guided roblem olving -. ollinear points lie on the same line.. nswers ma var. (ome people might note that the -coordinate of two of the points is the same so that the third point must have the same -coordinate to be collinear. ince it does not, the points are not collinear.). horizontal. o.. o.. ll points must have the same -coordinate, -.. o.. a, b ractice - Front ight. true. false. false. false.,. an three of the following pairs: and ; and ; and ; and ; and ; and ; and * * ) ) * * ) * ) ) ; and ; and ; and ; and ; and *GF ) * * ) * ) ) * * ) ) * ) * * ) * ) * ) KG JH * ) *H ) * ) JK * ) ) HG * ) *FG ) * ) *GK ) F JH F HG J HG FK JK H JK J FG H FK J KG KF JH. planes and. planes and ll-in-ne nswers Version Geometr

165 Geometr: ll-in-ne nswers Version (continued). ample: ) ) ) ) ) G 0. F and or G and., F. es F ) ). ample: ractice -. an three of the following: &, &M, &M, &. &. &. & &, &, &F, &F, &F 0. &F, &. &, & U. ample: Guided roblem olving -. pposite ras are two collinear ras with the same endpoint.. a line. ee graph in ercise answer.. nswers ma var. ample: (0,0) (nswers will be coordinates (, ), where =, <.) (0,0). es. (, ) Q (,) (,) ractice ,. no Guided roblem olving - W.. =. egment ddition ostulate.. ince =, = ().. = () =. =. = =. nswers ma var.. =, =, = Guided roblem olving -. ngle ddition ostulate. supplementar angles. m&q + m&q = 0. ( + ) + ( + 0) = 0. =.. m&q = ; m&q =. he sum of the angle measures should be 0; m&q + m&q = + = 0. a. = b. m& = ; m& = ractice X Y X X 0 Y X Y Geometr ll-in-ne nswers Version

166 Geometr: ll-in-ne nswers Version (continued). ractice ?? X W. true 0. false. false. true Guided roblem olving -. & > &. complementar angles. &. m& = m& =. m& = m& + m& = + =. m& + m& = 0 m& + = 0 m& =. m&f = m& =. nswers ma var. ample: he sum of the measures of the complementar angles should be 0 and the sum of the measures of the supplementar angles should be 0.. m&=, m&f =, m& =, and m& = X... (, ). (-, ) 0. (-0.,.). (, -). es; = = = =. " Q... (., ) Guided roblem olving -. istance Formula. he distance d between two points (, ) to (, ) is d Í ( ) ( ). o; the differences are opposites but the squares of the differences are the same.. XY Í ( ()) ( ). o the nearest tenth, XY =. units.. o the nearest tenth, XZ =.0 units.. Z is closer to X.. he results are the same; e.g., XY Í ( ) ( ()) Í, or about. units, as before.. YZ Í ( ()) ( ) Í; to the nearest tenth, YZ =. units. o the nearest tenth, XY+YZ+XZ =. units. ll-in-ne nswers Version Geometr

167 ractice -. in... m. p..p. cm; cm. in.; in.. 0 m; m. ;. 0; 0..p. 0,000p.. ; Guided roblem olving -. si. It is a two-dimensional pattern ou can fold to form a three-dimensional object.. rectangles.. 0 in.. he are equal.. 0 in.. nswers will var. ample: ( ) + ( ) + ( ); the results are the same, 0 in.. ( ) = in. : Graphic rganizer. ools of Geometr. nswers ma var. ample: patterns and inductive reasoning; measuring segments and angles; basic constructions; and the coordinate plane. heck students work. : eading omprehension. nswer ma var. ample: m/jf + m/fjk = 0, /HK > /KM,. oints, M, and are collinear.. and intersect at point M.. a : eading/writing Math mbols. ine is parallel to line M.. ine. ine segment GH. a. he length of segment XY is greater than the length of segment.. M = XY. GH = (K).. plane plane XYZ 0. : Visual Vocabular ractice. parallel planes. egment ddition ostulate. supplementar angles. opposite ras. isometric drawing. perpendicular lines. foundation drawing. right angle. congruent sides : Vocabular heck et: two-dimensional pattern that ou can fold to form a three-dimensional figure. onjecture: conclusion reached using inductive reasoning. ollinear points: oints that lie on the same line. Midpoint: point that divides a line segment into two congruent segments. ostulate: n accepted statement of fact. F: Vocabular eview uzzle ' UV * ) * ), * HI ), ) ' * ) JK > M, J > KM, * F ) * GH ), * ) * ), Geometr: ll-in-ne nswers Version (continued) Geometr ll-in-ne nswers Version Y I F F G J Q V I X U F G I I G Z Y G V V V I U G I U M V G V I M K J I H K I X Y H V U I W Y J X W U I H V I V J G W K Z G K I M I K Q U I M X F W F U Z G I K X Y V J W U U I J G H G I F U Q V F U M X Q K Z K W I K H I F

168 Geometr: ll-in-ne nswers Version (continued) hapter ractice -. ample: It is :00 noon on a rain da.. ample:. If ou are strong, then ou drink milk.. If a rectangle is a square, then it has four sides the same length.. If =, then - = ; true.. If m is positive, then m is positive; true.. If lines are parallel, then their slopes are equal; true.. Hpothesis: If ou like to shop; conclusion: Visit igeon Forge outlets in ennessee.. If ou visit igeon Forge outlets, then ou like to shop. 0. rinking ustain makes ou train harder and run faster.. If ou drink ustain, then ou will train harder and run faster.. If ou train harder and run faster, then ou drink ustain. Guided roblem olving -. Hpothesis: is an integer divisible b.. onclusion: is an integer divisible b.. Yes, it is true. ince is a factor of, it must be a factor of? =.. If is an integer divisible b then is an integer divisible b.. he converse is false. ountereamples ma var. et =. hen Í, which is not an integer and is not divisible b.. o. he conditional is true, so there is no such countereample.. o. definition, a general statement is false if a countereample can be provided.. If + =, then =. he original statement and the converse are both true. ractice -. wo angles have the same measure if and onl if the are congruent.. he converse, If n =, then n =, is not true.. If a whole number is a multiple of, then its last digit is either 0 or. If a whole number has a last digit of 0 or, then it is a multiple of.. If two lines are perpendicular, then the lines form four right angles. If two lines form four right angles, then the lines are perpendicular.. ample: ther vehicles, such as trucks, fit this description.. ample: aseball also fits this definition.. ample: leasing, smooth, and rigid all are too vague.. es. no 0. es Guided roblem olving -. good definition is clearl understood, precise, and reversible.. & and &, & and &. o. he are not supplementar.. linear pair has a common verte, shares a common side, and is supplementar.. es. es; es. linear pairs: & and &, & and &; not linear pairs: & and &, & and &, & and &, & and & ractice -. Football practice is canceled for Monda.. F is a right triangle.. If two lines are not parallel, then the intersect at a point.. If ou vacation at the beach, then ou like Florida.. amika lives in ebraska.. not possible. It is not freezing outside.. hannon lives in the smallest state in the United tates. Guided roblem olving -. conditional; hpothesis. Yes. eth will go.. nita, eth, isha, amon. o; onl two students went.. eth, isha, amon; no onl two went.. isha, amon. he answer is reasonable. It is not possible for another pair to go to the concert.. amon ractice -. U = M. =. J. ddition; ubtraction ropert of qualit; Multiplication ropert of qualit; ivision ropert of qualit. ubstitution. ubstitution. mmetric ropert of ongruence. efinition of omplementar ngles; 0, ubstitution;, implif;,, ubtraction ropert of qualit;, ivision ropert of qualit Guided roblem olving -. ngle ddition ostulate. ubstitution ropert of qualit; implif; ddition ropert of qualit; ivision ropert of qualit. 0. es; es. ; ractice m& = ; m& =. m& = 0; m& = 0. m&m = ; m&mq = ; m&qm =. m&w = m&w, m&w + m&w = 0; m&w + m&w = 0; m&w = m&w Guided roblem olving ee graph in ercise answer.. on the positive -ais. nswers ma var. can be an point on the positive -ais. ample: (0, ).. (0,) (0,0) (,) (, ) X(,0). he are adjacent complementar angles.. nswers ma var. can be an point on the line,. 0. ample: (, -).. a right angle. Yes; their sum corresponds to the right angle formed b the positive -ais and the positive -ais. 0. nswers ma var. can be an point on the negative -ais, sample: (-, 0) ll-in-ne nswers Version Geometr

169 Geometr: ll-in-ne nswers Version (continued) : Graphic rganizer. easoning and roof. nswers ma var. ample: conditional statements; writing biconditionals; converses; and using the aw of etachment. heck students work. : eading omprehension. degrees. degrees. b : eading/writing Math mbols. egment M is congruent to segment Q.. If p, then q.. he length of M is equal to the length of Q.. ngle XQV is congruent to angle.. If q, then p.. he measure of angle XQV is equal to the measure of angle.. p if and onl if q.. a b. = M 0. m/xyz = m/. b a. > M. a b. /XYZ / hapter ractice -. corresponding angles. alternate interior angles. sameside interior angles. and, and, and, and. and, and. and, and. m = 00, alternate interior angles; m = 00, corresponding angles or vertical angles. m =, corresponding angles; m =, vertical angles. = 0;, 0 0. = 0;, 0 : Visual Vocabular ractice. aw of etachment. hpothesis. istributive ropert. efleive ropert. aw of llogism. biconditional. conclusion. good definition. mmetric ropert : Vocabular heck ruth value: rue or false according to whether the statement is true or false, respectivel Hpothesis: he part that follows if in an if-then statement. iconditional: he combination of a conditional statement and its converse; it contains the words if and onl if. onclusion: he part of an if-then statement that follows then. onditional: n if-then statement. F: Vocabular eview uzzle U J I G J U I U G M U I G H Y H I I V I Guided roblem olving - I I. he top and bottom sides are parallel, and the left and right sides are parallel.. he two diagonals are transversals, and also each side of the parallelogram is a transversal for the two sides adjacent to it.. orresponding angles, interior and eterior angles are formed.. v, w and ; the lternate Interior ngles heorem, v =, w = and =.. nswers ma var. ossible answer: the ame-ide Interior ngles heorem, (w + ) + ( + ) = 0. ince w =, =. (he two s are equal b heorem -.). w =, =, v =, = ; es. v =, w =, =, = Geometr ll-in-ne nswers Version

170 Geometr: ll-in-ne nswers Version (continued) ractice -. l and m, onverse of ame-ide Interior ngles heorem. none. and, onverse of ame-ide Interior ngles heorem. H and I, onverse of orresponding ngles ostulate Guided roblem olving -. and m. transversals.. the angles measuring º and º. 0º. º. 0 - = or + = 0. =. With =, = and =. 0. = ractice -. rue. ver avenue will be parallel to Founders venue, and therefore ever avenue will be perpendicular to enter it oulevard, and therefore ever avenue will be perpendicular to an street that is parallel to enter it oulevard.. rue. he fact that one intersection is perpendicular, plus the fact that ever street belongs to one of two groups of parallel streets, is enough to guarantee that all intersections are perpendicular.. ot necessaril true. If there are more than three avenues and more than three boulevards, there will be some blocks bordered b neither enter it oulevard nor Founders venue.. a e. a e. a e. a e. If the number of statements is even, then n. If it is odd, then n. Guided roblem olving -. supplementar angles. right angle. n one of the following: ostulate -, or heorem -, -, - or It is congruent; ostulate a c. It is true for an line parallel to b.. Yes. he point is that a transversal cannot be perpendicular to just one of two parallel lines. It has to be perpendicular to both, or else to neither. ractice = ; = ; z =. v = ; w = ; t =. 0.. m = ; m = 0. right scalene. obtuse isosceles. equiangular equilateral Guided roblem olving -. three. 0. right triangle. z = 0; ecause it is given in the figure that.. heorem -, the riangle ngle-um heorem. =. =. # is a --0 right triangle. # is a --0 right triangle.. 0. # is a -- acute triangle.. Yes, all three are acute angles, with & visibl larger than & and &.. & ractice -. = 0; = 0. n =. a = 0; b = F. F and. Guided roblem olving -. theater stage, consisting of a large platform surrounding a smaller platform. he shapes in the bottom part of the figure ma represent a ramp for actors to enter and eit.. he measures of angles and. ; octagon. ( - )0 = 00 degrees... Yes, angle is an obtuse angle and angle is an acute angle.. trapezoids. 0 ractice -. = -. =- +. = -. = = +. = - 0. =- -. =- + =-; =. = ; = 0. (, 0) (0, ) ll-in-ne nswers Version Geometr

171 Geometr: ll-in-ne nswers Version (continued).. Guided roblem olving -. (0, ) (, 0) (, 0) (0, ). m. - = m(- ). lope of å å = ; slope of = -. he absolute values of the slopes are the same, but one slope is positive and the other is negative.. oint-slope form: - 0 = ( - 0); slope-intercept form: =. oint-slope form: - =- ( - ) or - 0 =- ( - ); å slope-intercept form: = f line : (0, 0); å of line : (0, 0). # appears to be an isosceles triangle, which is consistent with a horizontal base and two remiaining sides having slopes of equal magnitude and opposite sign.. lope = 0; = 0; -intercept = (0, 0) just as for line å (the intersect on the -ais). ractice -. neither;,? -. perpendicular;? - =-. parallel; - = -. perpendicular; = is a horizontal line, = 0 is a vertical line. perpendicular; -? =-. neither; -,? -. neither;,? -. parallel; =. = 0. = - Guided roblem olving -.. a right angle. m m =-. sides GH and GK. lope of GH = ; slope of GK =. roduct =- -. ides GH and GK are not perpendicular.. #GHK has no pair of perpendicular sides. It is not a right triangle.. o. &HGK; approimatel 0 0. lope of M = and slope of =-. he product of the slopes is -, so M and are perpendicular. ractice ample: K Q K b a G H Geometr ll-in-ne nswers Version

172 Geometr: ll-in-ne nswers Version (continued).. he appear to be both congruent and parallel.. he answers to tep are confirmed.. es; a parallelogram b b b : Graphic rganizer. arallel and erpendicular ines. nswers ma var. ample: properties of parallel lines; finding the measures of angles in triangles; classifing polgons; and graphing lines. heck students work. b : eading omprehension. spaces. spaces. 0. corresponding. $000; $0. the width of the stalls, 0 ft. b. Guided roblem olving -. a line segment of length c. onstruct a quadrilateral with one pair of parallel sides of length c, and then eamine the other pair.. he procedure is given on p. of the tet.. djust the compass to eactl span line segment c, end to end. hen tighten down the compass adjustment as necessar.. m. a c m l l m l : eading/writing Math mbols å å. m # n. m/ + m/ = 0.. m/m + m/mq = 0. / /F. ine is parallel to line.. he measure of angle is equal to the measure of angle XYZ.. ine is perpendicular to line F.. ngle and angle are complementar. 0. ngle is a right angle, or the measure of angle is 0.. ample answer: å å G, m/f = m/gf : Visual Vocabular ractice/high-use cademic Words. propert. conclusion. describe. formula. measure. approimate. compare. contradiction. pattern : Vocabular heck ransversal: line that intersects two coplanar lines in two points. lternate interior angles: onadjacent interior angles that lie on opposite sides of the transversal. ame-side interior angles: Interior angles that lie on the same side of a transversal between two lines. orresponding angles: ngles that lie on the same side of a transversal between two lines, in corresponding positions. Flow proof: convincing argument that uses deductive reasoning, in which arrows show the logical connections between the statements. F: Vocabular eview F. K. H. 0. G. I. J ll-in-ne nswers Version Geometr

173 Geometr: ll-in-ne nswers Version (continued) hapter ractice -. m = 0; m = 0. m = 0; m =. J,, J. J,,. Yes; GHJ IHJ b heorem - and b the efleive ropert of. herefore, GHJ IHJ b the definition of triangles.. o; Q V because vertical angles are congruent, and Q V b heorem -, but none of the sides are necessaril congruent. a. Given b. Vertical angles are. c. heorem - d. Given e. efinition of triangles Guided roblem olving -. right triangles. m& =, m& = m& = 0, and = in.. # is congruent to #KM means corresponding sides and angles are congruent.. and t.. m&k = m&m =. =. =. 0. t =. t =. he angle measures indicate that the two triangles are isosceles right triangles. his matches the appearance of the figure.. m&m = 0 ractice -. b. not possible. U XWV b. not possible. GHF b. Q b.. and. and 0. a. Given b. efleive ropert of ongruence c. ostulate Guided roblem olving -. I and bisects &I.. rove whatever additional facts can be proven about #I and #, based on the given information.. I >.. &I &.. #I # b ostulate -, the ide-ngle- ide()ostulate. It does not matter. he ide-ngle- ide ostulate applies whether or not the are collinear.. It does follow, because #I # and because I and Q are corresponding parts. ractice -. not possible. ostulate. heorem. ostulate. not possible. heorem. tatements easons. K M, K M. Given. JK M. Vertical s are.. JK M. ostulate. F. KHJ HKG or KJH HGK Guided roblem olving -. orresponding angles and alternate interior angles.. & and &.. & and &.. & and &. & > &, >, and & > &.. # > # b ostulate -, the ngle- ide-ngle () ostulate. Yes; heorem -, the ngle-ngle-ide () heorem; & > &.. o, because now there is no wa to demonstrate a second pair of congruent sides, nor a second pair of congruent angles. ractice -. is a common side, so b, and b.. FH is a common side, so FH HFG b, and H FG b.. Q is a common side, so Q Q b. Q Q b.. ZY and are vertical angles, so YZ b. Z b.. JKH and KM are vertical angles, so HJK MK b, and JK K b.. is a common side, so Q b, and Q b.. First, show that and are vertical angles. hen, show b. ast, show b. Guided roblem olving -. compass with a fied setting was used to draw two circular arcs, both centered at point but crossing in different locations, which were labeled and. he compass was used again, with a wider setting, to draw two intersecting circular arcs, one centered at and one at. he point at å which the new arcs intersected was labeled. Finall, line was drawn.. Find equal lengths or distances and eplain å wh is perpendicular to.. # and #. = and =.. # > #,b ostulate -, the ide-ide-ide () ostulate. & > & b. ince & > &, m& = m& and m& + m& = 0, it follows that m& = m& = 0.. from the definition of perpendicular and the fact that m& = m& = 0. he distances do not matter, so long as = and =. hat is what is required in order that # > #. 0. raw a line, and use the construction technique of the problem to construct a second line perpendicular to the first. hen do the same thing again to construct a third line perpendicular to the second line. he first and third lines will be parallel, b heorem -0. ractice -. = ; =. = 0; = 0. t = 0. = ; = 0; z =. =. z = 0. ; F. KJ; KIJ KJI. ; J J = 0; = Guided roblem olving -. ne angle is obtuse. he other two angles are acute and congruent.. Highlight an obtuse isosceles angle and find its angle measures, then find all the other angle measures represented in the figure. 0 Geometr ll-in-ne nswers Version

174 Geometr: ll-in-ne nswers Version (continued). ossible answer:. 0. 0, because the measure of each base angle is half the measure of an angle of the equilateral triangle.. 0 because the sum of the angles of the highlighted triangle must equal 0.. he other measures are 0 and 0. amples:. Yes; 0 = 0.. nswers ma var. ractice -. tatements easons. #, # F. Given., are right s.. erpendicular lines form right s.. F,. Given. F. H heorem. MJ and MJK are right s. erpendicular lines form right s. M MK Given MJ MJ efleive ropert of MJ MJK H heorem. VW. none. m and m F = 0. UV or V U. m and m X = 0. GI # JH Guided roblem olving -. wo congruent right triangles. ach one has a leg and a hpotenuse labeled with a variable epression.. the values of and for which the triangles are congruent b H. he two shorter legs are congruent.. = +. he hpotenuses are congruent.. + =. = ; = hpotenuse. shorter leg = ; es, this matches the figure.. he solution remains the same: = and =. he reason is that one is still solving the same two equations, = + and + =. ractice -. ZWX YXW;. M;. F G;. F G G F common side: FG. common angle: M K H J. ample: tatements easons. X Y. Given. X #, Y #. Given. m X and. erpendicular lines m Y = 0 form right s... efleive ropert of. Y X. ostulate Guided roblem olving -. he figure, a list of parallel and perpendicular pairs of sides, and one known angle measure, namel m& =. nine. he are congruent and have equal measures.. m& = m& = m& =. m& = 0. m& =. m& =. m& =. m&fg = 0 0. m& =. m& =, m& =, and m& =. m& = + =. m&fi = 0 -(m& + m&) = 0; m&h = m& = 0; m&fj = 0 - m& = ; m&ig = m&fi = 0 : Graphic rganizer. ongruent riangles. nswers ma var. ample: congruent figures; triangle congruence b,,, and ; proving parts of triangles congruent; the Isosceles riangle heorem. heck students work. : eading omprehension. Yes. Using the Isosceles riangle heorem, /W /Y. It is given that WX YX and WU YV. herefore nwux nyvx b.. here is not enough information. You need to know if, if / /,or if / /.. a ll-in-ne nswers Version Geometr

175 Geometr: ll-in-ne nswers Version (continued) : eading/writing Math mbols. ngle-ngle-ide. triangle XYZ. angle Q. line segment. line. ra WX. hpotenuseleg. line. angle 0. ngle-ide-ngle : Visual Vocabular ractice. theorem. congruent polgons. base angle of an isosceles triangle.. postulate. verte angle of an isosceles triangle. corollar. base of an isosceles triangle. legs of an isosceles triangle : Vocabular heck ngle: Formed b two ras with the same endpoint. ongruent angles: ngles that have the same measure. ongruent segments: egments that have the same length. orresponding polgons: olgons that have corresponding sides congruent and corresponding angles congruent. : n abbreviation for corresponding parts of congruent triangles are congruent. F: Vocabular eview uzzle. postulate. hpotenuse. angle. verte. side. leg. perpendicular. polgon. supplementar 0. parallel. corresponding hapter ractice - a. cm b. cm c. cm a.. cm b.. cm c.. cm..... a. b.. YZ, Q XZ, XY Q Guided roblem olving -. 0 units. he three sides of the large triangle are each bisected b intersections with the two line segments ling in the interior of the large triangle.. the value of. he are called midsegments.. he are parallel, and the side labeled 0 is half the length of the side labeled.. = 0. Yes; the side labeled appears to be about twice as long as the side labeled 0.. o. hose lengths are not fied b the given information. (he triangle could be verticall stretched or shrunk without changing the lengths of the labeled sides.) ll one can sa is that the midsegment is half as long as the side it is parallel to. ractice -. WY is the perpendicular bisector of XZ.... right triangle... ) isosceles triangle right triangle. J is the bisector of J..... right isosceles triangle Guided roblem olving -. <. ee answer to tep, above.. ee answer to tep, above.. lot a point and eplain wh it lies on the bisector of the angle at the origin.. line : ; line m: = 0. (0, ). = = ; es. heorem -, the onverse of the ngle isector heorem. m& = m& < 0. raw, m, and, then draw. ince = = 0, it follows that # > #, b H. hen > and & > & b. ractice -. (-, ). (, 0). altitude. median. perpendicular bisector. angle bisector a. (, 0) b. (-, -) a. ( 0, 0) b. (, -) Guided roblem olving - m. the figure and a proof with some parts left blank. Fill in the blanks... heorem -, the erpendicular isector heorem. ; X. the ransitive ropert of qualit. erpendicular isector. (his converse is heorem -.). he point of the proof is to demonstrate that n runs through point X. It would not be appropriate to show that fact as alread given in the figure.. othing essential would change. oint X would lie outside # (below å ), but the proof woud run just the same. ractice -. I and III. I and II. he angle measure is not.. ina does not have her driver s license.. he figure does not have eight sides.. is congruent to XYZ. a. If ou do not live in oronto, then ou do not live in anada; false. b. If ou do not live in anada, then ou do not live in oronto; true.. ssume that m m.. ssume that M does not intersect. 0. ssume that it is not sunn outside.. ssume that m 0. his means that m + m 0. his, in turn, means that the sum of the angles of eceeds 0, which contradicts the riangle ngle-um heorem. o the assumption that m 0 must be incorrect. herefore, m 0. Guided roblem olving -. Ice is forming on the sidewalk in front of oni s house.. Use indirect reasoning to show that the temperature of the sidewalk surface must be F or lower.. he temperature Geometr ll-in-ne nswers Version

176 Geometr: ll-in-ne nswers Version (continued) of the sidewalk in front of oni s house is greater than F.. Water is liquid (ice does not form) above F.. here is no ice forming on the sidewalk in front of oni s house.. he result from step contradicts the information identified as given in step.. he temperature of the sidewalk in front of oni s house is less than or equal to F.. If the temperature is above F, water remains liquid. his is reliabl true. onverse: If water remains liquid, the temperature is above F. his is not reliabl true. dding salt will cause water to remain liquid even below F.. uppose two people are each the world s tallest person. all them person and person. hen person would be taller than everone else, including, but b the same token would be taller than. It is a contradiction for two people each to be taller than the other. o it is impossible for two people each to be the World s allest erson. ractice -. M,.,.,.,. es; +, +, +. no; + 0. es; +. es; +, +, +. no; no;. +...,,.,,.,,.,, J Guided roblem olving -.. he side opposite the larger included angle is greater than the side opposite the smaller included angle.. he angle opposite the larger side is greater than the angle opposite the smaller side. he opposite sides each have a length of nearl the sum of the other two side lengths.. he opposite sides are the same length. he are corresponding parts of triangles that are congruent b. : Graphic rganizer. elationships Within riangles. nswers ma var. ample: midsegments of triangles; bisectors in triangles; concurrent lines, medians, and altitudes; and inverses, contrapositives, and indirect reasoning. heck students work. : eading omprehension. he width of the tar pit is 0 meters.. b : eading/writing Math mbols.. F.. G.. I. M. H. 0. K.... J. : Visual Vocabular ractice. median. negation. circumcenter. contrapositive. centroid. equivalent statements. incenter. inverse. altitude : Vocabular heck Midpoint: point that divides a line segment into two congruent segments. Midsegment of a triangle: he segment that joins the midpoints of two sides of a triangle. roof: convincing argument that uses deductive reasoning. oordinate proof: proof in which a figure is drawn on a coordinate plane and the formulas for slope, midpoint, and distance are used to prove properties of the figure. istance from a point to a line: he length of the perpendicular segment from the point to the line. F: Vocabular eview. altitude. line. median. negation. contrapositive. incenter. orthocenter. slope-intercept. eterior 0. obtuse. alternate interior. centroid. equivalent. right. parallel hapter ractice -. parallelogram. rectangle. quadrilateral. kite, quadrilateral. trapezoid, isosceles trapezoid, quadrilateral. square, rectangle, parallelogram, rhombus, quadrilateral. = ; = = = =. f = ; g = ; FG = GH = HI = IF =. parallelogram 0. kite Guided roblem olving -. a labeled figure, which shows an isosceles trapezoid. he nonparallel sides are congruent.. the measures of the angles and the lengths of the sides. m&g = c. c + (c - 0) = m& = m&g = 0, m& = m&f = 0. a - =. 0. = FG =, F =, G = = 0 = ( - )0. m& = m&g =, m& = m&f = ractice ; 0; 0. ; ; 0. ; ; 0. ; 0;. ; ; 0... ll-in-ne nswers Version Geometr

177 Geometr: ll-in-ne nswers Version (continued) Guided roblem olving -. the ratio of two different angle measures in a parallelogram. he consecutive angles are supplementar... the measures of the angles. he angles are supplementar angles, because the are consecutive.. + = 0. and. o; the lengths of the sides are irrelevant in this problem.. 0 and 0 ractice -. no. es. es. es. = ; =. = ; = 0. = ; the figure is a because both pairs of opposite sides are congruent.. = ; the figure is a because the congruent opposite sides are b the onverse of the lternate Interior ngles heorem.. o; the congruent opposite sides do not have to be. 0. o; the figure could be a trapezoid.. Yes; both pairs of opposite sides are congruent.. Yes; both pairs of opposite sides are b the converse of the lternate Interior ngles heorem.. o; onl one pair of opposite angles is congruent.. Yes; one pair of opposite sides is both congruent and. Guided roblem olving -. a labeled figure, which shows a quadrilateral that appears to be a parallelogram. he consecutive angles are supplementar.. find values for and which make the quadrilateral a parallelogram. m& + m& = 0, so that & and & meet the requirements for same-side interior angles on the transversal of two parallel lines (heorems - and -).. & > &, b heorem = 0; + =. =, =. m& = m& = and m& = m& =, which matches the appearance of the figure.. ( + 0) + ( + ) = 0; es ractice - a. rhombus b. ; ; ; a. rectangle b. ; ; 0; a. rectangle b. 0; 0; 0; 0 a. rhombus b. ; ; ; 0. ossible; opposite angles are congruent in a parallelogram.. Impossible; if the diagonals are perpendicular, then the parallelogram should be a rhombus, but the sides are not of equal length.. = ; HJ = ; IK =. = ; HJ = ; IK = a. 0; 0; ; b. cm 0a. ; 0; 0; 0b. 0 m Guided roblem olving -. labeled figure, which shows a parallelogram. ne angle is a right angle, and two adjacent sides are congruent. lgebraic epressions are given for the lengths of three line segments.. diagonals. Find the values of and.. It is a square. heorem - and the fact that > impl that all four sides are congruent. heorems - and - plus the fact that m& = 0 impl that all four angles are right angles.. congruent; bisect. - + = ( - ) + ( + ); - = +. = ; =. It was not necessar to know >, but it was necessar to know m& = 0. he ke fact, which enables the use of heorem - in addition to heorem -, is that is a rectangle. It does not matter whether all four sides are congruent.. 0 ractice -. ;. ;. ;. 0;. =. =. 0.; 0.. ;. 0; ; 0. 0; 0. =. = Guided roblem olving -. Isosceles trapezoid with >. & > & and & > &. > is Given. > because opposite sides of a parallelogram are congruent (heorem -). > is from the ransitive ropert of ongruenc.. Isosceles; >; because base angles of an isosceles triangle are congruent.. & > & because corresponding angles on a transversal of two parallel lines are congruent.. & > & b the ransitive ropert of ongruenc.. & is a same-side interior angle with &, and & is a same-side interior angle with &.. his is not a problem, because for > there is a similar proof with a line segment drawn from to a point ling on.. he two drawn segments can be shown to be congruent, and then one has two congruent right triangles b the H heorem. & > & follows b and & > & because the are supplements of congruent angles. ractice -. (.a, b); a. (0.a, 0); a. (0.a, b); "a b b b a. - a. (a, b); I(a, 0) 0. (a, b); I(a, 0). (-a, b). (-b, 0) Guided roblem olving -. a rhombus with coordinates given for two vertices. rhombus is a parallelogram with four congruent sides.. the coordinates of the other two vertices. he are the diagonals.. he bisect each other.. W(-r, 0), Z(0, -t). o; neither heorem - nor an other theorem or result would appl.. lope of WX = 0 and slope of YZ is undefined. his confirms heorem -0, which sas that the diagonals of a rhombus are perpendicular. ractice - p p a. q b. = m + b; q = q (p) +b; b = q - q ; p p = +q- p q q c. = r + p d. = q (r + p) + p q - rp p ;= + p + q - rp q q q q ;= q + q; intersection rp r at (r + p, q + q) e. q f. (r, q) g. = m + b; r r q = (r) + b; b = q - r r ;= + q - q q q q r r r rp r h. = (r + p) + q - ;= + + q - q q q q q ; p Geometr ll-in-ne nswers Version

178 Geometr: ll-in-ne nswers Version (continued) rp rp = q + q; intersection at (r + p, q + q) rp i. (r +p, q +q) a. (-a, 0) b. (-a, b) a b b c. (-, ) d. a Guided roblem olving -. kite FG with = F with the midpoint of each side identified. kite is a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent.. he midpoints are the vertices of a rectangle.. (-b,c), G(0, 0). (b, a + c), M(b, c), (-b, c), K(-b, a + c). lope of K = slope of M = 0, slopes of K and M are undefined. pposite sides are parallel; it is a rectangle.. djacent sides are perpendicular.. right angles 0. nswers will var. ample: a =, b =, c = ields the points (-, ), (0, ), F(, ), G(0, 0) with midpoints at (-, ), (-, ), (, ), and (, ). onnecting these midpoints forms a rectangle.. onstruct F and G. lope of F = 0, so F is horizontal. lope of G is undefined, so G is vertical. : Graphic rganizer. Quadrilaterals. nswers ma var. ample: classifing quadrilaterals; properties of parallelograms; proving that a quadrilateral is a parallelogram; and special parallelograms. heck students work. : eading omprehension. Q >, Q >, Q, Q V. o, it cannot be proven that nqv > nu because with H M U the given information, onl one side and one angle of the two triangles can be proven to be congruent. nother side or angle is needed. If it were given that QUV is a parallelogram, then the proof could be made.. ll four sides are congruent.. Yes. ince G > G b the efleive ropert, nfg > nhg b.. b : eading/writing Math mbols. H, H, or H. G, FG, or G. G.. rhombus. rectangle. square. isosceles trapezoid : Visual Vocabular ractice/high-use dademic Words. solve. deduce. equivalent. indirect. equal. analsis. identif. convert. common : Visual Vocabular heck onsecutive angles: ngles of a polgon that share a common side. Kite: quadrilateral with two pairs of congruent adjacent sides and no opposite sides congruent. arallelogram: quadrilateral with two pairs of parallel sides. hombus: parallelogram with four congruent sides. rapezoid: quadrilateral with eactl one pair of parallel sides. F: Vocabular eview uzzle G Z I Q U G 0 H M I G M Y I I U M U ll-in-ne nswers Version Geometr

179 Geometr: ll-in-ne nswers Version (continued) hapter ractice -. :. ft b 0 ft. ft b ft. 0 ft b ft. true. false. true. false. true 0. true :. :. 0. Guided roblem olving -. ratios.. the denominator,000,000 or,000,000.. ross-roduct ropert. 0.0,000,000, es.. ractice -. XYZ, with similarit ratio :. ot similar; corresponding sides are not proportional.. ot similar; corresponding angles are not congruent.. KM, with similarit ratio :. I..... ft 0.. cm..... Guided roblem olving -. equal....;...;.. no.. no..0;.. ince the quotients are not equal, the ratios are not equal, and the bills are not similar rectangles...0 ractice -. X XQ because vertical angles are. (Given). herefore X XQ b the M X XM ostulate.. ecause W = W = =, MX W b the heorem.. QM M QM M because vertical s are. hen, because M = M =, QM M b the heorem.. ecause X X X = X and X = X, X = X. X X because vertical angles are. herefore X X b 0 the heorem ft Guided roblem olving -. no; /. es; W,. It is a trapezoid.. he are congruent.. he are parallel.. he are congruent.. #Z and #WZ. ~ or ngle-ngle imilarit ostulate. o; there is onl one pair of congruent angles. 0. es; parallelogram, rhombus, rectangle, and square Guided roblem olving -. #, #, #.. ;... ;.. ". " 0. es. no (his would require the thagorean heorem.) ractice -... J.... = ; =.. = ; = Guided roblem olving -. parallel. ;. ;. 0. es. es. he sides would not be parallel.. he are similar triangles. : Graphic rganizer. imilarit. nswers ma var. ample: ratios and proportions; similar polgons; proving triangles similar; and similarit in right triangles. heck students work. : eading omprehension. H. similarit postulate he triangles have two similar angles.. :. :.. 0 ft. a H : eading/writing Math mbols. no. no. es. no. es. no. es,. es, Hpotenuse-eg heorem. es, or or 0. not possible : Visual Vocabular ractice. ngle-ngle imilarit ostulate. golden ratio. ide-ide-ide imilarit heorem. geometric mean. scale. ross-roduct ropert. golden rectangle. simplest radical form. ide-ngle-ide imilarit heorem : Vocabular heck imilarit ratio: he ratio of lengths of corresponding sides of similar polgons. ross-roduct ropert: he product of the etremes of a proportion is equal to the product of the means. atio: comparison of two quantities b division. Golden rectangle: rectangle that can be divided into a square and a rectangle that is similar to the original rectangle. cale: he ratio of an length in a scale drawing to the corresponding actual length. ractice ". ". h.. a. c. 0. = ; = ". = "; = ". " in. F: Vocabular eview... I.. J. K... G 0... M. H.. F Geometr ll-in-ne nswers Version

180 Geometr: ll-in-ne nswers Version (continued) hapter ractice -. ". ". ". ". in.. ft. cm. m. acute 0. obtuse. right Guided roblem olving -. the sum of the lengths of the sides. thagorean heorem. cm. cm cm. c = +.. cm. perimeter of rectangle = cm; es. nswers will var; eample: raw a cm cm grid, cop the given figure, measure the lengths with a ruler, add them together cm ractice -. = ; = ". ". ".. = ; = ". ". cm. 0. ft, ft. a = ; b = 0. p = ";q = ";r = ; s = " Guided roblem olving triangle. l. h.. Í or Í!.. or Í. ft. 0. min 0. es!. ft ractice -. tan = ; tan F =. tan = ; tan F = Guided roblem olving m& = m&x. 0. base: 0 cm, height: 0 cm es. ractice - 0 cm "0. sin = ; cos =. sin = ; cos = ". sin = ; cos =. sin = ; cos = Guided roblem olving - 0 cm. he sides are parallel.. sine. sin 0 w..0. es. cosine. cos.. nswers ma var. ample: cos 0 ; sin 0..,. ractice - a. angle of depression from the plane to the person b. angle of elevation from the person to the plane c. angle of depression from the person to the sailboat d. angle of elevation from the sailboat to the person.. ft.. ft.. ft.. d a. b. ft Guided roblem olving -. &e =, &d =&. congruent. m&e = m&d. - = ( + ).... es., ractice -..0,.0.., mi/h;.º north of east.. m;.0º south of west. º north of east. º west of north a., b. a., - b.. ample: W 0 ft ll-in-ne nswers Version Geometr

181 Guided roblem olving -. heck students work cos 0 ; 00 sin 0..; 0.., 0.., -0.., 0. ; due east. es 0. 00; due east : Graphic rganizer. ight riangles and rigonometr. nswers ma var. ample: the thagorean heorem; special right triangles; the tangent ratio; sine and cosine ratios; angles of elevation and depression; vectors. heck students work. : eading omprehension.. J.. J... b : eading/writing Math mbols. F. G..... H.. 0. #, #XYZ. m& <. : Visual Vocabular ractice triangle. inverse of tangent. congruent sides. tangent. thagorean heorem. hpotenuse triangle. thagorean triple. obtuse triangle : Vocabular heck btuse triangle: triangle with one angle whose measure is between 0 and 0. Isosceles triangle: triangle that has at least two congruent sides. Hpotenuse: he side opposite the right angle in a right triangle. ight triangle: triangle that contains one right angle. thagorean triple: set of three nonzero whole numbers a, b, and c that satisf the equation a + b = c. F: Vocabular eview uzzle tan Z sin 00 ; 00 Geometr: ll-in-ne nswers Version (continued) Geometr ll-in-ne nswers Version J H I Z H H Y M Y K W I Y F H V M M G G I U X Z I U I G V U Z K X I U I U F U F U Q X F V X G I G I M F M U M G Y Y M Q Y J Z V M H G M I M M F I J H Q M X M I J I V V K G I Y H G

182 Geometr: ll-in-ne nswers Version (continued) hapter ractice -. o; the triangles are not the same size.. Yes; the ovals are the same shape and size. a. and F b. and, and, F and F, F and F. (, ) ( -, - ). (, ) ( +, - ). (, ) ( +, + ). W (-, ), X (-, ), Y (, ), Z (, ). J (-, 0), K (-, ), (-, -). (, ) ( +, - ) 0. (, ) (,) ( +, + ) a. (-, -) b. (0, ), (-, ), Q (, ) Guided roblem olving -. the four vertices of a preimage and one of the vertices of the image. Graph the image and preimage.. (, ) and (0, 0). =, =, + a = 0, + b = 0. a =-; b =-; (, ) ( -, - ). (-, ), (, ), (-, ).. es. (, ) ( +, - ), (, ), (, ), (, 0) ractice -. (-, -). (-, -). (-, -). (, -). (, -). (, -) a. b. X Z W Y Z Y X W.. 0. (-, ) Guided roblem olving -. a point at the origin, and two reflection lines. reflection is an isometr in which a figure and its image have opposite orientations.. the image after two successive reflections... = (0,0) = (0,0) = (0,0) "(0, ) '(0,) '(0,). (0, - ). Yes. he -coordinate remains 0 throughout.. (0, 0) and (0, ) ll-in-ne nswers Version Geometr

183 Geometr: ll-in-ne nswers Version (continued) ractice -. I. I. I. GH. G.... slope of ; slope of ; slope of ; the slopes of perpendicular line segments are negative reciprocals.. square. es. (, ), (, -), (-, -) ractice -. he helmet has reflectional smmetr.. he teapot has reflectional smmetr.. he hat has both rotational and reflectional smmetr... Q Q. Guided roblem olving -. he coordinates of point, and three rotation transformations. It is assumed that the rotations are counterclockwise.. arallelogram, rhombus, square. slope of. ' Q' Q ' line smmetr and rotational smmetr line smmetr and 0 rotational smmetr line smmetr and rotational smmetr line smmetr line smmetr and rotational smmetr 0 rotational smmetr K 0 Geometr ll-in-ne nswers Version

184 Geometr: ll-in-ne nswers Version (continued). Guided roblem olving -. the coordinates of one verte of a figure that is smmetric about the -ais. ine smmetr is the tpe of smmetr for which there is a reflection that maps a figure onto itself.. the coordinates of another verte of the figure. images (and preimages). reflection across the -ais. (-, ). Yes. (-, ) ractice es. no. no.. H X. (-, -), Q (-, 0), (0, -) 0. (-, ), Q (-, 0), (0, ) Guided roblem olving -. description of a square projected onto a screen b an overhead projector, including the square s area and the scale factor in relation to the square on the transparenc.. he scale factor of a dilation is the number that describes the size change from an original figure to its image.. the area of the square on the transparenc. smaller; he scale factor >, so the dilation is an enlargement.. ;. ; eing as high and as wide, the square on the tranparenc has times the area.. ft. he shape does not matter. egardless of the shape, the figure is being enlarged b a factor of in two directions, so that the screen image has = as large an area as the figure on the transparenc.. 0 ft ractice -. I. II. III. IV J J m m m J. reflection. rotation. glide reflection 0. translation m m Guided roblem olving -. assorted triangles and a set of coordinate aes. a transformation. the transformation that maps one ll-in-ne nswers Version Geometr

185 Geometr: ll-in-ne nswers Version (continued) triangle onto another. he are congruent, which confirms that the could be related b one of the isometries listed. orresponding vertices: and, and Q, and M.. counterclockwise; clockwise. he triangles have opposite orientations, so the isometr must be a reflection or glide reflection.. o; there is no reflection line that will work.. a glide reflection consisting of a translation (, ) ( +, ) followed b a reflection across line = 0 (the -ais). Yes, both are horizontal.. 0º rotation with center Q, 0 ractice -... ossible answer:. heck students work.. nswers ma var. ample:..... amples:... es. no. es translational smmetr line smmetr, rotational smmetr, translational smmetr, glide reflectional smmetr line smmetr, rotational smmetr, translational smmetr, glide reflectional smmetr rotational smmetr, translational smmetr Guided roblem olving -. two different-sized equilateral triangles and a trapezoid. tessellation is a repeating pattern of figures that completel covers a plane, without gaps or overlaps.. heorem - sas that ever quadrilateral tessellates. If the three polgons can be joined to form a quadrilateral (using each polgon at least once), that quadrilateral can be the basis of a tessellation. : Graphic rganizer. ransformations. nswers ma var. ample: reflections; translations; compositions of reflections; and smmetr. heck students work. : eading omprehension..,,, and 0. he center of the count is the center of the square mile bounded b roads,, K and... 0 miles. a : eading/writing Math mbols. reflection; opposite. rotation; same. glide reflection; opposite. translation; same. rotation; same. translation; same : Visual Vocabular ractice/high-use cademic Words. theorem. smbols. reduce. application. simplif. eplain. characteristic. table. investigate : Vocabular heck ransformation: change in the position, size, or shape of a geometric figure. reimage: he given figure before a transformation is applied. Image: he resulting figure after a transformation is applied. Isometr: transformation in which an original figure and its image are congruent. omposition: transformation in which a second transformation is performed on the image of the first transformation. Geometr ll-in-ne nswers Version

186 Geometr: ll-in-ne nswers Version (continued) F: Vocabular eview uzzle M hapter 0 U 0 I Y M M M G Y H I Y F Y I H I I I I I U U I I F M V ractice Guided roblem olving 0-.. heck students work; (, ). a right angle bh.. å.... a square b ractice 0-. cm. in.. 0 m. ft. 0 cm.. in... square units.. in. Guided roblem olving 0-.. ; ;.. no. es. h(b b ) ractice 0-. = ; =.". = 0; p = ";q =. m = ; m = ; m = 0; m = 0. m = 0; m = 0; m = 0. 0 in.. in. I X Guided roblem olving he angles are all congruent and the sides are all congruent bisects ; ; ractice 0-. : ; :. : ; :. :. :. in. 00. cm. :. :. 0 ft Guided roblem olving 0-. scale. heck students work.. nswers will var. et a be the side opposite the 0 angle, b be the side opposite the 0 angle, and c be the side corresponding to 00 d. hen 00 d a < 0mm, b < 0mm and c < 0mm.. 0 mm d mm. < 0 mm. < mm. < 0 mm es.,000 ractice 0-.. cm. ft. 0. mm.. in... m. 0. mi.. km.. in... mm 0..0 in... m..0 ft.. ft. m ll-in-ne nswers Version Geometr

187 Geometr: ll-in-ne nswers Version (continued) Guided roblem olving 0-. subtract. 0. ap tan 0.. es 0. ractice 0-0. p..p. amples: F, F. amples: F, F. amples: F, p in.. p cm Guided roblem olving 0-. ;. ; ;. circumference; ractice 0-. p... p. p -.. p. p. p. p. p 0. p.. Guided roblem olving 0-. m a piece from 0? pr the outer ring. es. a piece from the top tier ractice 0-..%. 0.%..% %..%. 0% 0. 0%. 0%. % Guided roblem olving 0-. outcomes. event. F. FG. G. ; heck students work.. or about %. or about %; es. or 0% 0: Graphic rganizer. rea. nswers ma var. ample: areas of parallelograms and triangles; areas of trapezoids, rhombuses, and kites; areas of regular polgons; perimeters and areas of similar figures; trigonometr and area; circles and arcs; areas of circles and sectors; geometric probabilit. heck students work. 0: eading omprehension.. J.. G.. I... H 0. F. a + b = c 0 å å. m 0. M 0 0. ". #JK. XY > Q. ~. =. b 0: eading/writing Math mbols. 0 ft. ft. bundles. squares. $.00. $.00. minutes.. hours. $ 0. $ 0: Visual Vocabular ractice. area of a triangle. radius. altitude of a parallelogram. area of a trapezoid. area of a kite or rhombus. similarit ratio. apothem. perimeter (of an octagon). height of a triangle 0: Vocabular heck emicircle: Half a circle. djacent arcs: rcs on the same circle with eactl one point in common. pothem: he distance from the center to the side of a regular polgon. ircumference: he distance around a circle. oncentric circles: ircles that lie on the same plane and have the same center. 0F: Vocabular eview uzzle. adjacent arcs. semicircle. center. arc length. base. circle. central angle. congruent arcs. diameter 0. major arc. concentric circles. minor arc. thagorean triple. radius. apothem. sector. height. circumference Highness, there is no roal road to geometr. hapter ractice triangle. rectangle 0. rectangle Guided roblem olving -. a two-dimensional network. Verif uler s Formula for the network shown... ;. + = + ; his is true.. F becomes, V stas at, becomes. uler s Formula becomes + = +, which is still true.. + = + ; his is true. ractice -.. cm.. cm..0 ft.. cm a. 0 m b. 0 m a. in. b. in. a. cm b. 0 cm a. 00 m b. 00 m..p cm 0. 0p ft Guided roblem olving -. a picture of a bo, with dimensions. the amount of cardboard the bo is made of, in square inches. front and a back, each in. in.; a top and a bottom, each in. in.; and one narrow side, in. in.. Front and back: each 0 in. ; top and bottom: each in. ; one narrow side: in.. in.. in. is about half a square foot ( ft = in. ), which is reasonable. he answer is an Geometr ll-in-ne nswers Version

188 Geometr: ll-in-ne nswers Version (continued) approimation because the solution did not consider the overlap of cardboard at the glued joints, nor does it factor in the trapezoid-shaped cutout in the front and back sides.. in. ractice -. 0 cm. ft. p cm. 00p m. 0. m.. ft.. cm.. ft Guided roblem olving -. a solid consisting of a clinder and a cone. the surface area of the solid. he top of the solid is a circular base of the clinder, = pr. he side of the solid is the lateral area of the clinder, = prh. he bottom of the solid is the lateral area of the cone, = pr.. r = ft; h = ft; / " ft. op: =. ft ; side: =. ft ; bottom: 0. ft. ft. bout 0., or a little less than half. his seems reasonable.. ft ractice -..0 m.. in... in... cm. 0 in.. ft. ft. in.. ft 0. m Guided roblem olving -. dimensions for a laer of topsoil, and two options for buing the soil. which purchasing option is cheaper. ft. 00 ft. bags; $ ft < d. $ (or slightl less if the topsoil volume is not rounded up to the net whole number). buing in bulk. he two costs are quite close. It makes sense that buing in bulk would be slightl cheaper for a good-sized backard. 0. uing top soil b the bag would be less epensive because its would cost $ and in bulk would cost at least $.. ractice -., cm. 00 in.. 0,0 in.. 00 d.. in... m.. in m Guided roblem olving -. a description of a plumb bob, with dimensions. the plumb bob s volume. equilateral; cm. a = " cm; = " cm. rism: V = " cm ; pramid: V = " cm ; plum bob: V < cm. ; this seems reasonable. cm ractice -..0 in..,,. m.. cm.. m.. mi.,0. cm.. m.. cm. cm 0. cm Guided roblem olving -. the diameter of a hailstone, along with its densit compared to normal ice. the hailstone s weight. V.. in... in. pr.. lb. he densit of normal ice. his was given for comparison purposes onl oz ractice -. :. :. es; :. not similar. cm. in.. ft. 00 cm Guided roblem olving -. two different sizes for an image on a balloon, and the volume of air in the balloon for one of those sizes. the volume of air for the other size. he two sizes of the clown face have a ratio of :, or :.. :. 0 in. = in.. Yes, the two clown face heights are lengths, measured in inches (not in. or in. ).. cm : Graphic rganizer. urface rea and Volume. nswers ma var. ample: space figures and nets; space figures and drawing; surface areas; and volumes.. heck students work. : eading omprehension. rea = base times height; area of a triangle. slope = the difference of the -coordinates divided b the difference of the -coordinates; slope of the line between points. a squared plus b squared equals c squared, the thagorean heorem; the lengths of the sides of a right triangle. area = pi times radius squared; area of a circle. volume = pi times radius squared times height; volume of a clinder. the sum of the -coordinates of two points divided b, and the sum of the -coordinates of two points divided b ; midpoint of a line segment with endpoints (, ) and (, ). circumference = two times pi times the radius; circumference of a circle. volume = one-third pi times radius squared times height; volume of a cone. distance equals the square root of the quantit of the sum of the square of the difference of the -coordinates, squared, plus the difference of the -coordinates, squared; the distance between two points (, ) and (, ) 0. area = one-half the height times the sum of base one plus base two; area of a trapezoid. a : eading/writing Math mbols.. in. ft : Visual Vocabular ractice. prism. cross section. sphere. clinder. cone. pramid. polhedron. altitude. base ll-in-ne nswers Version Geometr

189 Geometr: ll-in-ne nswers Version (continued) : Vocabular heck Face: surface of a polhedron. dge: n intersection of two faces of a polhedron. ltitude: For a prism or clinder, a perpendicular segment that joins the planes of the bases. urface area: For a prism, clinder, pramid, or cone, the sum of the lateral area and the areas of the bases. Volume: measure of the space a figure occupies. F: Vocabular eview. polhedron. edge. verte. cross section. prism. altitude. right prism. slant height. cone 0. volume. sphere. hemispheres. similar figures. hpotenuse hapter ractice ". ". circumscribed. inscribed. circumscribed Guided roblem olving -. he area of one is twice the area of the other.. inscribed; circumscribed... r. r. "r.r. r = (r ): ne area is double the other area. 0.es. he same: he area of one circle is twice the area of the other circle. ractice r = ; m.. r = ";m Q ; U. J; K. ecause congruent arcs have congruent chords, = =. hen, because an equilateral triangle is equiangular, m = m = m. Guided roblem olving -. perpendicular. bisects. ;. right. radius. thagorean heorem.. es. ; r ractice -. and ; and. and.. = ; = 0; z =. = 0; = = 0; = 00; z = 0 a. m = 0 b. m = 0 c. m = 0 d. m = 00 Guided roblem olving a diameter. heck students work.. heck students work.. heck students work.. heck students work.. 0. heck students work. ractice = ; = ; z =. = 0; = 0; z = 0. = ; =. = ; = ; z =. = 0; = Guided roblem olving -. tangent ; ; es; he sum of the measures of the arcs should be 0.. pposite angles are not supplementar. ractice -. (0, 0), r =. (-, -), r =. + =. ( - ) +( - ) =. ( + ) + ( - ) =. ( + ) + ( + ) =. + =. + ( - ) =. ( - ) + ( + ) = 0. + ( + 0) = 00. (0, ). (0, 0) (, ) (, 0) Geometr ll-in-ne nswers Version

190 Geometr: ll-in-ne nswers Version (continued) Guided roblem olving -. (, );.. he are perpendicular ; es. -intercept: a = ; -intercept: b = ractice a third line parallel to the two given lines and midwa between them. a sphere whose center is the given point and whose radius is the given distance. the points within, but not on, a circle of radius in. centered at the given point. 0. in. X. cm Y 0. in. Guided roblem olving -. a line. heck students work.. he are perpendicular the midpoint of Q. (, ). -. = - 0. (, 0); es. : Graphic rganizer. ircles. nswers ma var. ample: tangent lines; chords and arcs; inscribed angles; and angle measures and segment lengths. heck students work. : eading omprehension nswers ma var. ample answers:.,. F,,. H. H F,,. F,,.,. /. / 0.. /H. /G. /H. /. H. a : * ) eading/writing * ) Math mbols ' F.. is tangent to (G.. WX > ZX. ' XY. HM = : Visual Vocabular ractice. tangent to a circle. standard form of an equation of a circle. secant. chord. circumscribed about. intercepted arc. inscribed in. inscribed angle. locus : Vocabular heck Inscribed in: circle inside a polgon with the sides of the polgon tangent to the circle. hord: segment whose endpoints are on a circle. oint of tangenc: he single point of intersection of a tangent line and a circle. angent to a circle: line, segment, or ra in the plane of a circle that intersects the circle in eactl one point. ircumscribed about: circle outside a polgon with the vertices of the polgon on the circle. F: Vocabular eview.. F. G..... M. I 0... H. K. J ll-in-ne nswers Version Geometr

191 Geometr: ll-in-ne nswers Version ame lass ate ame lass ate esson - esson bjective Use inductive reasoning to make conjectures 00 trand: Geometr opic: Mathematical easoning atterns and Inductive easoning ample. Using Inductive easoning Make a conjecture about the sum of the cubes of the first counting numbers. Find the first few sums. otice that each sum is a perfect square and that the perfect squares form a pattern. Vocabular. Inductive reasoning is conjecture is a conclusion ou reach using inductive reasoning. countereample is an eample for which the conjecture is incorrect. ample. FindingandUsingaattern Find a pattern for the sequence. Use the pattern to find the net two terms in the sequence.,,,,... ach term is half the preceding term. he net two terms are and. Quick heck.. Find the net two terms in each sequence. a.,,,,,,,,,... b. Monda, uesda, Wednesda, hursda, Frida,... c. nswers ma var. ample: reasoning based on patterns ou observe.,,... Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Make isometric and orthographic drawings raw nets for three-dimensional figures Vocabular. n isometric drawing of a three-dimensional object shows a corner view of the figure drawn on isometric dot paper. n orthographic drawing is the top view, front view, and right-side view of a threedimensional figure. net is a two-dimensional pattern ou can fold to form a three-dimensional figure. ample. rthographic rawing Make an orthographic drawing of the isometric drawing at right. rthographic drawings flatten the depth of a figure. n orthographic drawing shows three views. ecause no edge of the isometric drawing is hidden in the top, front, and right views, all lines are solid. Front op ight 00 trand: Geometr opic: imension and hape Quick heck.. Make an orthographic drawing from this isometric drawing. Front op ight rawings, ets, and ther Models Front Front ight ight ( ) ( ) 00 0 ( ) ( ) he sum of the first two cubes equals the square of the sum of the first two counting numbers. he sum of the first three cubes equals the square of the sum of the first three counting numbers. his pattern continues for the fourth and fifth rows. o a conjecture might be that the sum of the cubes of the first counting numbers equals the square of the sum of the first counting numbers, or (... ). Quick heck.. Make a conjecture about the sum of the first odd numbers. Use our calculator to verif our conjecture. he sum of the first odd numbers is, or. ail otetaking Guide Geometr esson - ame lass ate ample. rawing a et raw a net for the figure with a square base and four isosceles triangle faces. abel the net with its dimensions. hink of the sides of the square base as hinges, and unfold the figure at these edges to form a net. he base of each of the four isosceles triangle faces is a side of the square. Write in the known dimensions. Quick heck.. he drawing shows one possible net for the Graham rackers bo. cm cm 0 cm cm GHM K cm 0 cm cm GHM K 0 cm raw a different net for this bo. how the dimensions in our diagram. nswers ma var. ample: cm 0 cm 0 cm cm cm Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

192 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Understand basic terms of geometr Understand basic postulates of geometr 00 trand: Geometr opic: imension and hape oints, ines, and lanes plane is a flat surface that has no thickness. wo points or lines are coplanar if the lie on the same plane. postulate or aiom is an accepted statement of fact. lane Vocabular and Ke oncepts. ostulate - hrough an two points there is eactl one line. ostulate - ine t is the onl line that passes through points and. If two lines intersect, then the intersect in eactl one point. ostulate - and intersect at. If two planes intersect, then the intersect in eactl one line. W ostulate - t * ) * ) lane and plane W intersect in. hrough an three noncollinear points there is eactl one plane. point is a location. pace is the set of all points. line is a series of points that etends in two opposite directions without end. ollinear points are points that lie on the same line. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Identif segments and ras ecognize parallel lines Vocabular. segment is the part of a line consisting of two egment endpoints and all points between them. ndpoint ndpoint ra is the part of a line consisting of one endpoint and ) a YX YX all the points of the line on one side of the endpoint. X Y ndpoint pposite ras are two collinear ras with the same endpoint. Q ) and ) Q are opposite ras. arallel lines are coplanar lines that do not intersect. kew lines are noncoplanar; therefore, the are not parallel and do not intersect. H G F is parallel and G are arallel planes are planes that do not intersect. G H lane is J I * ) egments, as, arallel ines and lanes 00 trand: Geometr opic: elationships mong Geometric Figures to F. skew lines. parallel to plane GHIJ. t amples. Identifing ollinear oints In the figure at right, X name three points that are collinear and three points that are not collinear. oints Y, Z, and W lie on a line, so the are collinear. m Z Y W Using ostulate - hade the plane that contains X, Y, and Z. Z oints X, Y, and Z are the vertices of one of the four triangular faces of the pramid. o shade the plane, shade the interior of Y the triangle formed b X, Y, and Z. X Quick heck.. Use the figure in ample. a. re points W, Y, and X collinear? b. ame line m in three different was. * ) * ) * ) nswers ma var. ample: ZW, WY, YZ.. a. hade plane VWX. b. ame a point that is coplanar Z with points V, W, and X. Y V no Y W X ail otetaking Guide Geometr esson - ame lass ate amples. aming egments and as ame the segments and ras in the figure. he labeled points in the figure are,, and. segment is a part of a line consisting of two endpoints and all points between them. segment is named b its two endpoints. o the segments are (or ) and (or ). ra is a part of a line consisting of one endpoint and all the points of the line on one side of that endpoint. ra is named b its endpoint first, followed b an other point on the ra. o the ras are ) and ). Identifing arallel lanes Identif a pair of parallel planes in our classroom. lanes are parallel if the do not intersect. If the walls of our classroom are vertical, opposite walls are parts of parallel planes. If the ceiling and floor of the classroom are level, the are parts of parallel planes. Quick heck. ) ). ritical hinking Use the figure in ample. and form a line. re the opposite ras? plain. o; the do not have the same endpoint.. Use the diagram to the right. a. ame three pairs of parallel planes. W QVU, U QVW, Q WVU * Q ) b. ame a line that is parallel to. * ) V c. ame a line that is parallel to * plane ) QUV. nswers ma var. ample: V W W U Q V Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

193 0 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Find the lengths of segments 00 trand: Measurement opic: Measuring hsical ttributes Measuring egments amples. Using the egment ddition ostulate If, find the value of. hen find and. Use the egment ddition ostulate (ostulate -) to write an equation. egment ddition ostulate Vocabular and Ke oncepts. 0 ostulate -: uler ostulate he points of a line can be put into one-to-one correspondence with the real numbers so that the distance between an two points is the absolute value of the difference of the corresponding numbers. ostulate -: egment ddition ostulate If three points,, and are collinear and is between and, then. coordinate is a point s distance and direction from zero on a number line. the length of u a b u a Q a coordinate of coordinate of ongruent ( ) segments are segments with the same length. cm cm midpoint is a point that divides a segment into two congruent segments. Geometr esson - ail otetaking Guide b midpoint ame lass ate esson - esson bjectives Find the measures of angles Identif special angle pairs Vocabular and Ke oncepts. 00 trand: Measurement opic: Measuring hsical ttributes Measuring ngles ostulate -: rotractor ostulate et and be opposite ras in a plane.,, and all the ras with endpoint that can * be drawn on one side of ) can be paired with the 0 real numbers from 0 to 0 so that a. is paired with 0 and is paired with 0. b. If is paired with and is paired with, ml u u then. 0 0 ostulate -: ngle ddition ostulate If point is in the interior of, then If is a straight angle, then m m m. m m 0. n angle ( ) is formed b two ras with the same endpoint. he ras are the sides of the angle and the endpoint is the verte of the angle. acute angle right angle obtuse angle 0,, 0 0 0,, 0 n acute angle has measurement between 0 and 0. right angle has a measurement of eactl 0. n obtuse angle has measurement between 0 and 0. straight angle has a measurement of eactl 0. ongruent angles are two angles with the same measure Q Q 0 0 straight angle ( ) ( ) ubstitute. implif the left side. ubtract from each side. ivide each side b. ( ) 0 ubstitute for. ( ) 0 and, which checks because the sum equals. Finding engths M is the midpoint of. Find M, M, and. M Use the definition of midpoint to write an equation. M M efinition of midpoint ubstitute. dd to each side. ubtract from each side. ivide each side b. M ( ) ubstitute for. M ( ) M M egment ddition ostulate M and M are each, which is half of, the length of. Quick heck.. G 00. Find the value of. hen find F and FG., F 0; FG 0. Z is the midpoint of XY, and XY. Find XZ F G ail otetaking Guide Geometr esson - ame lass ate amples. aming ngles ame the angle at right in four was. he name can be the number between the sides of the angle: l. he name can be the verte of the angle: lg. Finall, the name can be a point on one side, the verte, and a point on the other side of the angle: G lg or lg. Using the ngle ddition ostulate uppose that m and m. Find m. Use the ngle ddition ostulate (ostulate -) to solve. ngle ddition m m m ostulate m ubstitute for m and for m. m ubtract from each side. Quick heck.. a. ame two other was. l, l b. ritical hinking Would it be correct to name an of the angles? plain. o, angles have for a verte, so ou need more information in the name to distinguish them from one another.. If m G, find m GF. G F Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

194 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Use a compass and a straightedge to construct congruent segments and congruent angles Use a compass and a straightedge to bisect segments and angles asic onstructions 00 trand: Geometr opic: elationships mong Geometric Figures onstructing the erpendicular isector Given:.* ) * ) onstruct: XY so that XY at the midpoint M of. tep ut the compass point on point and draw a long arc. e sure that the opening is greater than. Vocabular. onstruction is using a straightedge and a compass to draw a geometric figure. straightedge is a ruler with no markings on it. compass is a geometric tool used to draw circles and parts of circles called arcs. erpendicular lines are two lines that intersect to form right angles. perpendicular bisector of a segment is a line, segment, or ra that is perpendicular to the segment at its midpoint, thereb bisecting the segment into two congruent segments. n angle bisector is a ra that divides an angle into two congruent coplanar angles. amples. onstructing ongruent egments onstruct W congruent K to KM. tep raw a ra with endpoint. tep pen the compass the length of KM. tep With the same compass setting, put the compass point on point. raw an arc that intersects the ra. abel the point of intersection W. KM W Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Find the distance between two points in the coordinate plane Find the coordinates of the midpoint of a segment in the coordinate plane Ke oncepts. Formula: he istance Formula he distance d between two points (, ) and (, ) is d Formula: he Midpoint Formula he coordinates of the midpoint M of with endpoints (, ) and (, ) are the following: amples. Finding the Midpoint has endpoints (, ) and (, ). Find the coordinates of its midpoint M. Use the Midpoint Formula. et (, ) be (, ) and (, ) be (, ). he midpoint has coordinates, ( ). ( ) he -coordinate is he -coordinate is ( ) ( ) ( ) M, ( ) 00 trand: Measurement opic: Measuring hsical ttributes.. Midpoint Formula J K W ubstitute for and for. implif. ubstitute for and for. implif. M he oordinate lane tep With the same compass setting, put the compass point on point and draw another long arc. abel the points where the two arcs intersect as X and Y. tep raw. he point of intersection of and is M, the midpoint of. * XY ) * XY ) at the midpoint of, so is the perpendicular bisector of. Quick heck.. Use a straightedge to draw XY. hen construct so that XY.. raw. onstruct its perpendicular bisector. * XY ) ail otetaking Guide Geometr esson - * XY ) X Y X X M ame lass ate Finding an ndpoint he midpoint of G is M(, ). ne endpoint is (, ). Find the coordinates of the other endpoint G. Use the Midpoint Formula. et (, ) be (, ) and the midpoint a, b be (, ). olve for and, the coordinates of G. Find the -coordinate of G. d Use the Midpoint Formula. d Multipl each side b. d implif. he coordinates of G are (, ). 0 Quick heck.. Find the coordinates of the midpoint of XY with endpoints X(, ) and Y(, ). (, ). he midpoint of XY has coordinates (, ). X has coordinates (, ). Find the coordinates of Y. (, ) Y Y he coordinates of the midpoint M are (, ). Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

195 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - erimeter, ircumference, and rea Finding rea of a ircle Find the area of in terms of π. esson bjectives Find perimeters of rectangles and squares, and circumferences of circles Find areas of rectangles, squares, and circles 00 trand: Measurement opic: Measuring hsical ttributes In, r. d. π r Formula for the area of a circle π (. ) ubstitute. for r.. π. d Ke oncepts. he area of is.p d. erimeter and rea s s quare with side length s. erimeter rea s s h ectangle with base b and height h. erimeter rea ostulate - If two figures are congruent, then their areas are equal. ostulate -0 he area of a region is the sum of the areas of its non-overlapping parts. amples. Finding ircumference G has a radius of. cm. Find the circumference of G in terms of π. hen find the circumference to the nearest tenth. p r Formula for circumference of a circle.. π( ) ubstitute for r. π act answer 0.00 Use a calculator. he circumference of G is p, or about 0. cm. b b bh d r ircle with radius r and diameter d. ircumference pd or pr rea pr Geometr esson - ail otetaking Guide h b h. cm G ame lass ate esson - esson bjectives ecognize conditional statements Write converses of conditional statements Vocabular and Ke oncepts. 00 trand: Geometr opic: Mathematical easoning onditional tatements onditional tatements and onverses tatement ample mbolic Form You read it onditional If an angle is a straight angle, p q If p, then its measure is 0. then q. onverse If the measure of an angle is q p If q, 0, then it is a straight angle. then p. conditional is an if-then statement. he hpothesis is the part that follows if in an if-then statement. he conclusion is the part of an if-then statement (conditional) that follows then. he truth value of a statement is true or false according to whether the statement is true or false, respectivel. he converse of the conditional if p, then q is the conditional if q, then p. amples. Identifing the Hpothesis and the onclusion Identif the hpothesis and conclusion: If two lines are parallel, then the lines are coplanar. In a conditional statement, the clause after if is the hpothesis and the clause after then is the conclusion. Hpothesis: wo lines are parallel. Quick heck.. a. Find the circumference of a circle with a radius of m in terms of π. p m b. Find the circumference of a circle with a diameter of m to the nearest tenth..m. You are designing a rectangular banner for the front of a museum. he banner will be ft wide and d high. How much material do ou need in square ards? d ail otetaking Guide Geometr esson - ame lass ate Writing the onverse of a onditional Write the converse of the following conditional. If, then. he converse of a conditional echanges the hpothesis and the conclusion. onditional onverse Hpothesis onclusion Hpothesis onclusion o the converse is: If, then. Quick heck.. Identif the hpothesis and the conclusion of this conditional statement: If, then. Hpothesis: onclusion:. Write the converse of the following conditional: If two lines are not parallel and do not intersect, then the are skew. If two lines are skew, then the are not parallel and do not intersect. onclusion: he lines are coplanar. 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

196 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Write biconditionals ecognize good definitions iconditionals and efinitions 00 trand: Geometr opics: imension and hape; Mathematical easoning Identifing a Good efinition Is the following statement a good definition? plain. n apple is a fruit that contains seeds. he statement is true as a description of an apple. change n apple and a fruit that contains seeds. he converse reads: fruit that contains seeds is an apple. Vocabular and Ke oncepts. iconditional tatements biconditional combines p q and q p as p q. tatement ample mbolic Form You read it iconditional n angle is a straight angle if p q p if and q and onl if its measure is 0. onl if. biconditional statement is the combination of a conditional statement and its converse. biconditional contains the words if and onl if. amples. Writing a iconditional onsider the true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. onditional: If, then 0. o write the converse, echange the hpothesis and conclusion. onverse: If 0, then. When ou subtract from each side to solve the equation, ou get. ecause both the conditional and its converse are true, ou can combine them in a biconditional using the phrase if and onl if. iconditional: if and onl if 0. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Use the aw of etachment Use the aw of llogism Vocabular and Ke oncepts. 00 trand: Geometr opic: Mathematical easoning aw of etachment If a conditional is true and its hpothesis is true, then its In smbolic form: If p q is a true statement and p is true, then q is true. aw of llogism If p q and q r are true statements, then p r is a true statement. is true. eductive reasoning is a process of reasoning logicall from given facts to a conclusion. amples. Using the aw of etachment gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make? he first sentence contains a conditional statement. he hpothesis is it rains. ecause the hpothesis is true, the gardener can conclude that the garden will be watered. conclusion Using the aw of etachment For the given statements, what can ou conclude? Given: If is acute, then m 0. is acute. conditional and its hpothesis are both given as true. the aw of etachment, ou can conclude that the conclusion of the conditional, m 0, is true. eductive easoning here are man fruits that contain seeds but are not apples, such as lemons and peaches. hese are countereamples, so the converse of the statement is false. he original statement is not a good definition because the statement is not reversible. Quick heck.. onsider the true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. onditional: If three points are collinear, then the lie on the same line. onverse: If three points lie on the same line, then the are collinear. he converse is true. iconditional: hree points are collinear if and onl if the lie on the same line.. Is the following statement a good definition? plain. square is a figure with four right angles. It is not a good definition because a rectangle has four right angles and is not necessaril a square. ail otetaking Guide Geometr esson - ame lass ate Using the aw of llogism Use the aw of llogism to draw a conclusion from the following true statements: If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle. he conclusion of the first conditional is the hpothesis of the second conditional. his means that ou can appl the aw of llogism. he aw of llogism: If p q and q r are true statements, then p r is a true statement. o ou can conclude: If a quadrilateral is a square, then it is a rectangle. Quick heck.. uppose that a mechanic begins work on a car and finds that the car will not start. an the mechanic conclude that the car has a dead batter? plain. o, there could be other things wrong with the car, such as a fault starter.. If a baseball plaer is a pitcher, then that plaer should not pitch a complete game two das in a row. Vladimir uñez is a pitcher. n Monda, he pitches a complete game. What can ou conclude? nswers ma var. ample: Vladimir uñez should not pitch a complete game on uesda.. If possible, state a conclusion using the aw of llogism. If it is not possible to use this law, eplain wh. a. If a number ends in 0, then it is divisible b 0. If a number is divisible b 0, then it is divisible b. If a number ends in 0, then it is divisible b. b. If a number ends in, then it is divisible b. If a number ends in, then it is divisible b. ot possible; the conclusion of one statement is not the hpothesis of the other statement. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

197 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective onnect reasoning in algebra and geometr Ke oncepts. easoning in lgebra 00 trand: lgebra and Geometr opics: lgebraic epresentations; Mathematical easoning amples. Justifing teps in olving an quation Justif each step used to solve for. Given: ddition ropert of qualit ubtraction ropert of qualit ivision ropert of qualit roperties of qualit ddition ropert If a b, then a c b c. ubtraction ropert If a b, then a c b c. Multiplication ropert If a b, then a c b c. a b ivision ropert If a b and c 0, then. c c efleive ropert a a. mmetric ropert If a b, then b a. ransitive ropert If a b and b c, then a c. ubstitution ropert If a b, then b can replace a in an epression. istributive ropert a(b c) ab ac. roperties of ongruence efleive ropert mmetric ropert If, then. If, then. ransitive ropert If and F, then F. If and, then. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives rove and appl theorems about angles Vocabular and Ke oncepts. roving ngles ongruent 00 trand: Geometr opic: elationships mong Geometric Figures heorem -: Vertical ngles heorem congruent Vertical angles are. and heorem -: ongruent upplements heorem If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. heorem -: ongruent omplements heorem If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. heorem - ll right angles are congruent. heorem - If two angles are congruent and supplementar, then each is a right angle. vertical angles and are vertical angles, as are and. Vertical angles are two angles whose sides form two pairs of opposite ras. adjacent angles and are adjacent angles, as are and. djacent angles are two coplanar angles that have a common side and a common verte but no common interior points. Using roperties of qualit and ongruence ame the propert that justifies each statement. If Q, Q, and, then. Use the ransitive ropert of ongruence for the first two parts of the hpothesis: If Q and Q, then. Use the ransitive ropert of ongruence for and the third part of the hpothesis: If and, then. Quick heck.. Fill in each missing reason. ) Given: M bisects K. M bisects K Given ngle m M m KM efinition of isector ubstitution ropert of qualit ubtraction ropert of qualit ivision ropert of qualit. ame the propert of equalit or congruence illustrated. a. XY XY efleive ropert of ongruence b. If m and m, then m m ransitive or ubstitution ropert of qualit M ( 0) K ail otetaking Guide Geometr esson - ame lass ate complementar angles and are complementar angles. and are supplementar angles. wo angles are complementar angles if wo angles are supplementar angles if the sum of their measures is 0. the sum of their measures is 0. theorem is a conjecture that is proven. paragraph proof is a convincing argument that uses deductive reasoning in which statements and reasons are connected in sentences. amples. Using the Vertical ngles heorem Find the value of. he angles with labeled measures are vertical angles. ppl the Vertical ngles heorem to find. ( ) Vertical ngles heorem ddition ropert of qualit ubtraction ropert of qualit ivision ropert of qualit ( 0) roving heorem - Write a paragraph proof of heorem - using the diagram at the right. tart with the given: and are supplementar, and are supplementar. the definition of supplementar angles, m m 0 and m m 0. substitution, m m m m. Using the ubtraction ropert of qualit, subtract m from each side. You get m m, or. Quick heck.. efer to the diagram for ample. a. Find the measures of the labeled pair of vertical angles. 0 b. Find the measures of the other pair of vertical angles. supplementar angles c. heck to see that adjacent angles are supplementar. 0 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

198 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Identif angles formed b two lines and a transversal rove and use properties of parallel lines Vocabular and Ke oncepts. ostulate -: orresponding ngles ostulate If a transversal intersects two parallel lines, then corresponding angles are congruent. heorem -: lternate Interior ngles heorem If a transversal intersects two parallel lines, then alternate interior angles are congruent. heorem -: ame-ide Interior ngles heorem If a transversal intersects two parallel lines, then same-side interior angles are supplementar. m m trand: Measurement opic: Measuring hsical ttributes transversal is a line that intersects two coplanar lines at two distinct points. roperties of arallel ines lternate interior angles are nonadjacent interior angles that lie on opposite sides of the transversal. ame-side interior angles are interior angles that lie on the same side of the transversal. orresponding angles are angles that lie on the same side of the transversal and in corresponding positions relative to the coplanar lines. t Geometr esson - ail otetaking Guide a b / m t t / m and are alternate interior angles. and are same-side interior angles. and corresponding angles. ame lass ate esson - esson bjectives Use a transversal in proving lines parallel Vocabular and Ke oncepts. 00 trand: Measurement opic: Measuring hsical ttributes ostulate -: onverse of the orresponding ngles ostulate If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel. heorem -: onverse of the lternate Interior ngles heorem If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. heorem -: onverse of the ame-ide Interior ngles heorem If two lines and a transversal form same-side interior angles that are supplementar, then the two lines are parallel. heorem -: onverse of the lternate terior ngles heorem If two lines and a transversal form alternate eterior angles that are congruent, then the two lines are parallel. heorem -: onverse of the ame-ide terior ngles heorem If two lines and a transversal form same-side eterior angles that are supplementar, then the two lines are parallel. flow proof uses arrows to show the logical connections between the statements. easons are written below the statements. roving ines arallel / m m. / m If, then m. If and are supplementar, then m. If, then m. If and are supplementar, then m. amples. ppling roperties of arallel ines In the diagram of afaette egional irport, the black segments are runwas and the gra areas are taiwas and terminal buildings. ompare and the angle vertical to. lassif the angles as alternate interior angles, same-side interior angles, or corresponding angles. he angle vertical to is between the runwa segments. is between the runwa segments and on the opposite side of the transversal runwa. ecause alternate interior angles are not adjacent and lie between the lines on opposite sides of the transversal, and the angle vertical to are alternate interior angles. Finding Measures of ngles In the diagram at right, m and p q. Find m and m. corresponding angles and the angle are. ecause m, m b the orresponding ngles ostulate. ecause and are adjacent angles that form a straight angle, m m 0 b the ngle ddition ostulate. If ou substitute for m, the equation becomes m 0. ubtract from each side to find m. Quick heck.. Use the diagram in ample. lassif and as alternate interior angles, same-side interior angles, or corresponding angles. same-side interior angles. Using the diagram in ample find the measure of each angle. Justif each answer. a. ; Vertical angles are congruent b. ; orresponding angles are congruent c. ; ame-side interior angles are supplementar ail otetaking Guide Geometr esson - p q ame lass ate amples. roving heorem - If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. Given: rove: m Write the flow proof below of the lternate Interior ngles heorem as a paragraph proof. Given / m ransitive ropert of? Vertical angles are congruent ongruence the Vertical ngles heorem,., so b the ransitive ropert of ongruence. ecause and are corresponding angles, m b the onverse of the orresponding ngles ostulate. Using ostulate - Use the diagram at the right.which lines, if an, must be parallel if and are supplementar? Justif our answer with a theorem or postulate. It is given that and are supplementar. he diagram shows that and are supplementar. ecause supplements of the same angle are congruent (ongruent upplements heorem),. ecause ) and are congruent corresponding angles, ) K b the onverse of the orresponding ngles ostulate. Quick heck.. uppl the missing reason in the flow proof from ample. If corresponding angles are congruent, then the lines are parallel.. Use the diagram from ample. Which lines, if an, must be parallel if? plain. / m K / m u K u ; onverse of orresponding ngles ostulate Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

199 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives elate parallel and perpendicular lines Ke oncepts. heorem - If two lines are the are parallel parallel to the same line to each other. 00 trand: Geometr opic: elationships mong Geometric Figures, then heorem -0 In a plane, if two lines are perpendicular to the same line, m n then the are parallel to each other. heorem - In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. n m arallel and erpendicular ines a b c Geometr esson - ail otetaking Guide n a b. t m n. ame lass ate esson - esson bjectives lassif triangles and find the measures of their angles Use eterior angles of triangles Vocabular and Ke oncepts. heorem -: riangle ngle-um heorem he sum of the measures of the angles of a triangle is 0. m m m 0 heorem -: riangle terior ngle heorem he measure of each eterior angle of a triangle equals the sum of the measures of its two remote interior angles. n acute right n obtuse n equiangular triangle has triangle has triangle has triangle has three acute angles. one right angle. one obtuse angle. three congruent angles. n equilateral n isosceles scalene triangle has triangle has triangle has three congruent sides. at least two congruent sides. no congruent sides. n eterior angle of a polgon is an angle formed b a side and an etension of an adjacent side. emote interior angles are the two nonadjacent interior angles corresponding to each eterior angle of a triangle. 00 trand: Geometr opic: elationships mong Geometric Figures terior angle emote interior angles m arallel ines and the riangle ngle-um heorem amples. roof of heorem -0 Use the diagram at the right. Which angle would ou use with to prove the theorem In a plane, if two lines are perpendicular to the same line, then the are parallel to each other (heorem -0) using the onverse of the lternate Interior ngles heorem instead of the onverse of the orresponding ngles ostulate? r s t the Vertical ngles heorem, is congruent to its vertical angle. ecause, is congruent to the vertical angle of b the ransitive ropert of ongruence. ecause alternate interior angles are congruent, ou can use the vertical angle of and the onverse of the lternate Interior ngles heorem to prove that the lines are parallel. Using lgebra Find the value of for which m. he labeled angles are alternate interior angles. If m, the alternate interior angles are congruent, ( ) / ( ) m and their measures are equal. Write and solve the equation. 0 dd to each side. 0 ubtract from each side. 0 ivide each side b. Quick heck.. ritical hinking In a plane, if two lines form congruent angles with a third line, must the lines be parallel? raw a diagram to support our answer. o; answers ma var. ample:. Find the value of for which a b. plain how ou can check our answer. ; (), and 0 ( ) ail otetaking Guide Geometr esson - ame lass ate amples. ppling the riangle ngle-um heorem Find m Z. m Z 0 riangle ngle-um heorem m Z 0 implif. m Z ubtract from each side. ppling the riangle terior ngle heorem plain what happens to the angle formed b the back of the chair and the armrest as ou make a lounge chair recline more. ack rm he eterior angle and the angle formed b the back of the chair and the armrest are adjacent angles, which together form a straight angle. s one measure increases, the other measure decreases. he angle formed b the back of the chair and the armrest increases as ou make a lounge chair recline more. Quick heck.. a. M is a right triangle. M is a right angle and m is. Find m. b. easoning plain wh this statement must be true: If a triangle is a right triangle, its acute angles are complementar. he sum of the measures of the angles of a triangle is 0. If ou subtract the measure of the right angle from 0, ou get 0. he sum of the measures of the other two angles is 0, so the are complementar.. a. Find m. 0 (0 0 ) (0 0 ) b. ritical hinking Is it true that if two acute angles of a triangle are complementar, then the triangle must be a right triangle? plain. rue, because the measures of the two complementar angles add to 0, leaving 0 for the third angle. a b X Z Y Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

200 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives lassif polgons Find the sums of the measures of the interior and eterior angles of polgons Vocabular and Ke oncepts. heorem -: olgon ngle-um heorem he olgon ngle-um heorems 00 trand: Geometr opic: elationships mong Geometric Figures n equilateral polgon has all sides congruent. n equiangular polgon has all angles congruent. regular polgon is both equilateral and equiangular. amples. lassifing olgons lassif the polgon at the right b its sides. Identif it as conve or concave. tarting with an side, count the number of sides clockwise around the figure. ecause the polgon has sides, it is a dodecagon. hink of the he sum of the measures of the angles of an n-gon is (n )0. heorem -: olgon terior ngle-um heorem he sum of the measures of the eterior angles of a polgon, one at each verte, is 0. For the pentagon, m m m m m 0. polgon is a closed plane figure with at least three sides that are segments. he sides intersect onl at their endpoints, and no two adjacent sides are collinear. polgon conve polgon does not have diagonal points outside of the polgon. concave polgon has at least one diagonal with points outside of the polgon. ot a polgon; not a closed figure Geometr esson - ail otetaking Guide ot a polgon; two sides intersect between endpoints. Y iagonal M W K G Q conve polgon concave polgon ame lass ate esson - esson bjectives Graph lines given their equations Write equations of lines 00 trand: lgebra opics: atterns, elations, and Functions; lgebraic epresentations ines in the oordinate lane Vocabular. lope-intercept form he slope-intercept form of a linear equation is m b. he standard form of a linear equation is. he point-slope form for a nonvertical line is tandard form m( ). ( ) oint-slope amples. Graphing ines Using Intercepts Use the -intercept and -intercept to graph 0. o find the -intercept, substitute o find the -intercept, substitute 0 for and solve for. 0 for and solve for. 0 0 ( 0 ) 0 ( 0 ) he -intercept is. he -intercept is. point on the line is (,0). point on the line is (0, ). lot (, 0) and (0, ). raw the line containing the two points. (, 0) form polgon as a star. raw a diagonal connecting two adjacent points of the star. hat diagonal lies outside the polgon, so the dodecagon is concave. Finding a olgon ngle um Find the sum of the measures of the angles of a decagon. decagon has 0 sides, so n 0. um (n )(0) olgon ngle-um heorem 0 0 ( )(0) ubstitute for n.? 0 ubtract. 0 implif. Quick heck.. lassif each polgon b its sides. Identif each as conve or concave. a. b. heagon; conve. Find the sum of the measures of the angles of a -gon. 0 octagon; concave ail otetaking Guide Geometr esson - ame lass ate Using oint-lope Form Write an equation in point-slope form of the line with slope that contains (, ). m ( ) Use point-slope form. ( ) implif. ( ) ( ) ubstitute for m and (, ) for (, ). Quick heck.. Graph each equation. a. b. c.. Write an equation of the line that contains the points (, 0) and Q(, ). 0 ( ) or ( ) (0, ) 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 Geometr ll-in-ne nswers Version

201 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives elate slope and parallel lines elate slope and perpendicular lines lopes of arallel and erpendicular ines 00 trand: Measurement opic: Measuring hsical ttributes Finding lopes for erpendicular ines Find the slope of a line perpendicular to. o find the slope of the given line, rewrite the equation in slope-intercept form. Ke oncepts. lopes of arallel ines ubtract from each side. ivide each side b. If two nonvertical lines are parallel, their slopes are equal. If the slopes of two distinct nonvertical lines are equal, the lines are parallel. n two vertical lines are parallel. lopes of erpendicular ines If two nonvertical lines are perpendicular, the product of their slopes is. If the slopes of two lines have a product of, the lines are perpendicular. n horizontal line and vertical line are perpendicular. amples. etermining Whether ines are arallel re the lines and parallel? plain. he equation is in slope-intercept form. Write the equation in slope-intercept form. ubtract from each side. ivide each side b. he line has slope. he line has slope. he lines are not parallel because their slopes are not equal. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives onstruct parallel lines onstruct perpendicular lines 00 trand: Geometr opic: elationships mong Geometric Figures ample. onstructing m amine the diagram at right. plain how to construct congruent to H. onstruct the angle. Use the method learned for constructing congruent angles. tep With the compass point on point H, draw an arc that intersects the sides of H. tep With the same compass setting, put the compass point on point. raw an arc. tep ut the compass point below point where the arc * ) intersects H. pen the compass to the length where the arc intersects line. Keeping the same compass setting, put the compass point above point where the * ) arc intersects H. raw an arc to locate a point. tep Use a straightedge to draw line m through the point ou located and point. Quick heck.. Use ample. plain wh lines and m must be parallel. If corresponding angles are congruent, the lines are parallel b the onverse of orresponding ngles ostulate. onstructing arallel and erpendicular ines H J m / he line has slope. Find the slope of a line perpendicular to. et m be the slope of the perpendicular line. m he product of the slopes of perpendicular lines is. m? Multipl each side b. ( ) m implif. Quick heck.. re the lines and parallel? plain. Yes; ach line has slope and the -intercepts are different.. Find the slope of a line perpendicular to 0. m ail otetaking Guide Geometr esson - ame lass ate ample. erpendicular From a oint to a ine amine the * ) construction. t what special point does G meet line? / oint is the same distance from point as it is from point F because the arc was made with one compass opening. oint G is the same distance from point as it is from point F because both arcs were made with the same compass opening. * ) his means that G intersects line at the midpoint of F, * ) and that G is the perpendicular bisector of F. Quick heck. * ) * ) * ) * ). Use a straightedge to draw F. onstruct FG so that FG F at point F. G F G F Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

202 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective ecognize congruent figures and their corresponding parts Vocabular and Ke oncepts. heorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. ongruent polgons are polgons that have corresponding sides congruent and corresponding angles congruent. amples. aming ongruent arts QJ. ist the congruent corresponding parts. ist the corresponding sides and angles in the same order. ngles: Q J ides: > Q J QJ 00 trand: Geometr opic: ransformation of hapes and reservation of roperties F Using ongruenc XYZ KM, m Y, and m M. Find m X. Use the riangle ngle-um heorem and the definition of congruent polgons to find m X. m X m Y m Z 0 riangle ngle-um heorem m Z m M orresponding angles of congruent triangles are congruent. m Z ubstitute for m M. m X 0 ubstitute. m X 0 implif. m X ubtract from each side. Geometr esson - ail otetaking Guide F ongruent Figures I F ame lass ate esson - esson bjective rove two triangles congruent using the and ostulates Ke oncepts. ostulate -: ide-ide-ide () ostulate If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are G congruent. ostulate -: ide-ngle-ide () ostulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. GHF Q F 00 trand: Geometr opic: ransformation of hapes and reservation of roperties amples. roving riangles ongruent Given: M is the midpoint of XY, X > Y rove: MX MY Write a paragraph proof. You are given that M is the midpoint of XY, and X Y. Midpoint M implies that MX MY. M > M b the efleive ropert of ongruence, so MX MY b the ostulate. H F X Q F Q M J H GHI G F riangle ongruence b and Y Finding ongruent riangles an ou conclude that? ist corresponding vertices in the same order. If, then. he diagram above shows, not. he statement is not true. otice that,, and. lso, and. Using heorem -, ou can conclude that. ince all of the corresponding sides and angles are congruent, the triangles are congruent. he correct wa to state this is. Quick heck.. WY MKV. ist the congruent corresponding parts. Use three letters for each angle. ides: WY MK W MV Y KV ngles: WY MVK lwy lvmk lwy lmkv. It is given that WY MKV. If m Y, what is m K? plain. m K orresponding angles of congruent triangles are congruent and have the same measure.. an ou conclude that JK M? Justif our answer. J K o. he corresponding sides are not necessaril equal. M ail otetaking Guide Geometr esson - ame lass ate Using >. What other information do ou need to prove b? It is given that >. lso, > b the efleive ropert of ongruence. You now have two pairs of corresponding congruent sides. herefore, if ou know, ou can prove b. Quick heck.. Given: HF HJ, FG JK, H is the midpoint of GK. rove: FGH JKH tatements easons. HF > HJ, FG > JK. Given. H is the midpoint of GK.. Given. GH > HK. efinition of Midpoint. FGH JKH. ostulate. What other information do ou need to prove b? F G H K J Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

203 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective rove two triangles congruent using the ostulate and the heorem Ke oncepts. 00 trand: Geometr opic: ransformation of hapes and reservation of roperties riangle ongruence b and ample. Writing a roof Write a two-column proof that uses. Given:, rove: tatements easons ostulate -: ngle-ide-ngle () ostulate.,. Given 0 If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. HG K G heorem -: ngle-ngle-ide () heorem If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. M XG G ample. Using uppose that F is congruent to and I is not congruent to. ame the triangles that are congruent b the ostulate. he diagram shows and F > > G. If F, then F G. herefore, FI G b. Quick heck.. Using onl the information in the diagram, can ou conclude that IF is congruent to either of the other two triangles? plain. o; nl one angle and one side are shown to be congruent. t least one more congruent side or angle is necessar to prove congruence with,, or. Geometr esson - ail otetaking Guide G K H ame lass ate esson - esson bjective Use triangle congruence and to prove that parts of two triangles are congruent Vocabular. stands for: orresponding arts of ongruent riangles are ongruent. amples. ongruence tatements he diagram shows the frame of an umbrella. What congruence statements besides can ou prove from the diagram, in which > and are given? b the efleive ropert of ongruence, and b. because corresponding parts of congruent triangles are congruent. When two triangles are congruent, ou can form congruence statements about three pairs of corresponding angles and three pairs of corresponding sides. ist the congruence statements. ides: > Given efleive ropert of ongruence ther congruence statement ngles: l l Given orresponding arts of ongruent riangles ther congruence statement 00 trand: Geometr opic: ransformation of hapes and reservation of roperties ib haft M X Using ongruent riangles: tretcher F I. l l. If lines are parallel, then. angles are congruent. alternate interior. >. efleive ropert of ongruence.. heorem Quick heck.. In ample, eplain how ou could prove using. l l. It is given that You know that l l b the l l lternate Interior ngles heorem. heorem -,. the efleive ropert of ongruence,. herefore, k k b. ail otetaking Guide Geometr esson - ame lass ate Using ight riangles ccording to legend, one of apoleon s followers used congruent triangles to estimate the width of a river. n the riverbank, the officer stood up straight and lowered the visor of his cap until the farthest thing he could see was the edge of the opposite bank. He then turned and noted the spot on his side of the river that was in line with his ee and the tip of his visor. F Given: G and F are right angles; G F. he officer then paced off the distance to this spot and declared that distance to be the width of the river! he given states that G and F are right angles. What conditions must hold for that to be true? G and F are the angles that the officer makes with the ground. o the officer must stand perpendicular to the ground, and the ground must be level or flat. Quick heck.. In ample, what can ou sa about and? plain. G he are congruent, because supplements of congruent angles are congruent.. ecall ample. bout how wide was the river if the officer paced off 0 paces and each pace was about feet long? 0 feet he congruence statements that remain to be proved are l l and. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

204 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective Use and appl properties of isosceles triangles Vocabular and Ke oncepts. heorem -: Isosceles riangle heorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. heorem -: onverse of Isosceles riangle heorem If two angles of a triangle are congruent, then the sides opposite those sides are congruent. heorem - he bisector of the verte angle of an isosceles triangle is the perpendicular bisector of the base. ' and bisects. he legs of an isosceles triangle are the congruent sides. he base of an isosceles triangle is the third, or non-congruent, side. Isosceles and quilateral riangles 00 trand: Geometr opic: elationships mong Geometric Figures Verte ngle egs ase ase ngles Geometr esson - ail otetaking Guide he verte angle of an isosceles triangle is formed b the two congruent sides (legs). he base angles of an isosceles triangle are formed b the base and the legs. ame lass ate esson - esson bjective rove triangles congruent using the H heorem Vocabular and Ke oncepts. heorem -: Hpotenuse-eg (H) heorem If the hpotenuse and a leg of one right triangle are congruent to the hpotenuse and leg of another right triangle, then the triangles are congruent. he hpotenuse of a right triangle is its longest side, or the side opposite the right angle. he legs of a right triangle are the two shortest sides, or the sides that are not opposite the right angle. amples. roving riangles ongruent ne student wrote M b the H heorem for the diagram. Is the student correct? plain. he diagram shows the following congruent parts. > M lm egs 00 trand: Geometr opic: ransformation of hapes and reservation of roperties Hpotenuse ince is the hpotenuse and is a leg of right triangle, and M is the hpotenuse and is a leg of right triangle M, the triangles are congruent b the H heorem. ongruence in ight riangles M ample. Using the Isosceles riangle heorems plain wh is isosceles. and X are alternate interior angles formed * b X,, and the transversal. ecause X ), X. he diagram shows that X. the ransitive ropert of ongruence,. You can use the onverse of the Isosceles riangle heorem to conclude that >. the definition of an isosceles triangle, is isosceles. Quick heck. a. In the figure, suppose m. Find m and m. b. In the figure, can ou deduce that X is isosceles? o; neither lx nor lx can be shown to be congruent to l. m ; m 0 ail otetaking Guide Geometr esson - ame lass ate wo-olumn roof Using the H heorem Given: and are right angles, > rove: tatements easons. and are. Given right angles. and. efinition of ight riangle are right triangles. >. Given.. efleive ropert of ongruence.. H Quick heck.. M Q Which two triangles are congruent b the H heorem? Write a correct congruence statement. M Q. You know that two legs of one right triangle are congruent to two legs of another right triangle. plain how to prove the triangles are congruent. ince all right angles are congruent, the triangles are congruent b. X he student is correct. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

205 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Identif congruent overlapping triangles rove two triangles congruent b first proving two other triangles congruent amples. Identifing ommon arts ame the parts of the sides that FG and HG share. Identif the overlapping triangles. H arts of sides G and G are shared b FG and HG. hese parts are HG and FG, respectivel. Using wo airs of riangles Write a paragraph proof. X Given: XW > YZ, XWZ and YZW are right angles. rove: XW YZ lan: XW YZ b if WXZ lzyw. hese angles are congruent b if XWZ YZW. hese triangles are congruent b. roof: You are given XW > YZ. ecause XWZ and YZW are W right angles, XWZ YZW. WZ > ZW b the efleive ropert of ongruence. herefore, XWZ YZW b. WXZ ZYW b, and XW YZ because vertical angles are congruent. herefore, XW YZ b. Using orresponding arts of ongruent riangles 00 trand: Geometr opic: elationships mong Geometric Figures Geometr esson - ail otetaking Guide ame lass ate esson - esson bjective Use properties of midsegments to solve problems Vocabular and Ke oncepts. heorem -: riangle Midsegment heorem If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length. midsegment of a triangle is a segment connecting the midpoints of two sides. coordinate proof is a form of proof in which coordinate geometr and algebra are used to prove a theorem. amples. Finding engths In XYZ, M,, and are midpoints. he perimeter of M is 0. Find and YZ. ecause the perimeter of M is 0, ou can find. M M 0 efinition of perimeter 0 ubstitute for M. 00 trand: Geometr opics: elationships mong Geometric Figures for M and 0 implif. ubtract from each side. Use the riangle Midsegment heorem to find YZ. M YZ riangle Midsegment heorem M Y G F Y Z Midsegments of riangles Z Quick heck.. he diagram shows triangles from the scaffolding that workers used when the repaired and cleaned the tatue of ibert. a. ame the common side in and. b. ame another pair of triangles that share a common side. ame the common side. nswers ma var. ample: and ;. Write a two-column proof. Given: >, Q Q rove: Q Q tatements easons., lq lq. Given. Q > Q. efleive ropert of ongruence. Q Q.. Q > Q. lq. Q.. Q > Q. efleive ropert of ongruence Q. Q. ail otetaking Guide Geometr esson - Q ame lass ate Identifing arallel egments Find m M and m M. M and are cut b transversal, so M and are corresponding angles. M b the riangle Midsegment heorem, so M b the orresponding ngles ostulate. m M because congruent angles have the same measure. In M, M, so m M m M b the Isosceles riangle heorem. m M b substitution. Quick heck.. 0 and. Find,, and. ; 0; 0. ritical hinking Find m VUZ. Justif our answer. X U Y V Z ; UVXY so VUZ and YXZ are corresponding and congruent. F M YZ ubstitute for M. YZ Multipl each side b. 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

206 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective Use properties of perpendicular bisectors and angle bisectors isectors in riangles 00 trand: Geometr opics: elationships mong Geometric Figures ample. Using the ngle isector heorem Find, F, and F in the diagram at right. F F ngle isector heorem Vocabular and Ke oncepts. ubstitute. dd to each side. F heorem -: erpendicular isector heorem ubtract from each side. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. heorem -: onverse of the erpendicular isector heorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. heorem -: ngle isector heorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. heorem -: onverse of the ngle isector heorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the angle bisector. he distance from a point to a line is the length of the perpendicular segment from the point to the line. 0 * ) * ) is in. from and. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjective Identif properties of perpendicular bisectors and angle bisectors Identif properties of medians and altitudes of a triangle Vocabular. 00 trand: Geometr opics: elationships mong Geometric Figures oncurrent lines are three or more lines that meet in one point. he point of concurrenc is the point at which concurrent lines intersect. circle is circumscribed about a polgon when the vertices ircumscribed circle of the polgon are on the circle. he circumcenter of a triangle is the point of Q concurrenc of the perpendicular bisectors of a triangle. Median median of a triangle is a segment that Q has as its endpoints a verte of the triangle and the midpoint of the opposite circumcenter side. amples. Finding the ircumcenter Find the center of the circle that circumscribes XYZ. ecause X has coordinates (, ) and Y has (, ) coordinates, XY lies on the vertical line. he perpendicular bisector of XY is the horizontal line that (, ) passes through (, ) or, so the equation of the perpendicular bisector of XY is. ecause X has coordinates (, ) and Z has (, ) coordinates, XZ lies on the horizontal line. he perpendicular bisector of XZ is the vertical line that passes through (, ) or (, ), so the equation of the perpendicular bisector of XZ is. oncurrent ines, Medians, and ltitudes Y X (, ) Z. ivide each side b. F (. ). ubstitute. F (. ). ubstitute. Quick heck. ) ) a. ccording to the diagram, how far is K from H? From? 0; 0 ) b. What ) can ou conclude about K? K is the angle bisector of lh. c. Find the value of. 0 d. Find m H. 0 K 0 ( 0) H ail otetaking Guide Geometr esson - ame lass ate Finding engths of Medians M is the centroid of W, and WM. W Find WX. he centroid is the point of concurrenc of the medians of Z Y a triangle. M he medians of a triangle are concurrent at a point that is X two-thirds the distance from each verte to the midpoint of the opposite side. (heorem -) ecause M is the centroid of W, WM WX. WM WX heorem - WX ubstitute for WM. WX Multipl each side b. Quick heck.. a. Find the center of the circle that ou can circumscribe about the triangle with vertices (0, 0), (, 0), and (0, ). (, ) b. ritical hinking In ample, eplain wh it is not necessar to find the third perpendicular bisector. heorem -: ll the perpendicular bisectors of the sides of a triangle are concurrent.. Using the diagram in ample, find MX. heck that WM MX WX. raw the lines and. he intersect at the point (, ). his point is the center of the circle that circumscribes XYZ. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

207 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Write the negation of a statement and the inverse and contrapositive of a conditional statement Use indirect reasoning Inverses, ontrapositives, and Indirect easoning 00 trand: Geometr opics: Mathematical easoning Writing the Inverse and ontrapositive Write the inverse and contrapositive of the conditional statement If is equilateral, then it is isosceles. o write the inverse of a conditional, negate both the hpothesis and the conclusion. Hpothesis onclusion onditional: If is equilateral, then it is isosceles. egate both. Vocabular and Ke oncepts. egation, Inverse, and ontrapositive tatements tatement ample mbolic Form You ead It onditional If an angle is a straight angle, p q If p, then q. then its measure is 0. egation (of p) n angle is not a straight angle. p ot p. Inverse If an angle is not a straight angle, p q If not p, then not q. then its measure is not 0. ontrapositive If an angle s measure is not 0, q p If not q, then not p. then it is not a straight angle. he negation of a statement has the opposite truth value from that of the original statement. he inverse of a conditional statement negates both the hpothesis and the conclusion. he contrapositive of a conditional switches the hpothesis and the conclusion and negates both. conditional contrapositive and its alwas have the same truth value. quivalent statements have the same truth value. amples. Writing the egation of a tatement Write the negation of is not a conve polgon. he negation of a statement has the opposite truth value. he negation of is not in the original statement removes the word not. he negation of is not a conve polgon is is a conve polgon. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Use inequalities involving angles of triangles Use inequalities involving sides of triangles Ke oncepts. 00 trand: Geometr opics: elationships mong Geometric Figures orollar to the riangle terior ngle heorem he measure of an eterior angle of a triangle is greater than the measure of each of its remote interior angles. m m and m m heorem -0 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. If XZ XY, then m Y m Z. heorem -: riangle Inequalit heorem he sum of the lengths of an two sides of a triangle is greater than the length of the third side. M M M M M M amples. ppling heorem -0 In GY, G, GY, and Y 0. ist the angles from largest to smallest. o two sides of GY are congruent, so the largest angle lies opposite the longest side. Find the angle opposite each side. Y he longest side is 0. he opposite angle, G, is the largest. he shortest side is. he opposite angle,, is the smallest. From largest to smallest, the angles are G, Y,. Inequalities in riangles X M G Y 0 Z Inverse: If is not equilateral, then it is not isosceles. o write the contrapositive of a conditional, switch the hpothesis and conclusion, then negate both. Hpothesis onclusion onditional: If is equilateral, then it is isosceles. witch and negate both. ontrapositive: If is not isosceles, then it is not equilateral. Quick heck.. Write the negation of each statement. a. m XYZ 0. he measure of XYZ is not more than 0. b. oda is not uesda. oda is uesda.. Write (a) the inverse and (b) the contrapositive of Maa ngelou s statement If ou don t stand for something, ou ll fall for anthing. a. If ou stand for something, ou won t fall for anthing. b. If ou won t fall for anthing, then ou stand for something. ail otetaking Guide Geometr esson - ame lass ate Using the riangle Inequalit heorem an a triangle have sides with the given lengths? plain. a. cm, cm, cm ccording to the riangle Inequalit heorem, the sum of the lengths of an two sides of a triangle is greater than the length of the third side. he sum of and is not greater than. o cannot, a triangle have sides with these lengths. b. in., in., in. Yes can, a triangle have sides with these lengths. Quick heck.. ist the angles of in order from smallest to largest.,,. an a triangle have sides with the given lengths? plain. a. m, m, and m no; is not greater than b. d, d, and d es;,, and ft ft ft Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

208 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective efine and classif special tpes of quadrilaterals Ke oncepts. 0 pecial Quadrilaterals Kite o pairs of parallel sides Isosceles rapezoid parallelogram is a quadrilateral with both pairs of opposite sides parallel. rhombus is a parallelogram with four congruent sides. Quadrilateral pair of parallel sides rapezoid rectangle is a parallelogram with four right angles. trapezoid is a quadrilateral with eactl one pair of parallel sides. 00 trand: Geometr opic: elationships mong Geometric Figures pairs of parallel sides ectangle kite is a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent. square is a parallelogram with four congruent sides and four right angles. n isosceles trapezoid is a trapezoid whose nonparallel sides are congruent. lassifing Quadrilaterals arallelogram quare hombus Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Use relationships among sides and among angles of parallelograms Use relationships involving diagonals of parallelograms and transversals Vocabular and Ke oncepts. heorem - pposite sides of a parallelogram are congruent. 00 trand: Geometr opic: elationships mong Geometric Figures heorem - pposite angles of a parallelogram are congruent. heorem - he diagonals of a parallelogram bisect each other. roperties of arallelograms amples. Using lgebra Find the value of in. hen find m. pposite angles of a ( ) parallelogram are congruent. dd to each side. 0 ubtract from each side. ( ) 0 ivide each side b. 0 0 m ubstitute for. m m 0 onsecutive angles of a parallelogram are supplementar. m 0 ubstitute for m. 0 m ubtract from each side. U ample. lassifing b oordinate Methods etermine the most precise name for the quadrilateral with vertices Q(, ), (, ), H(, ), and (0, ). Graph quadrilateral QH. Find the slope of each side. H lope of Q ( ) lope of H ( ) Q lope of H 0 lope of Q 0 H is parallel to Q because their slopes are equal. H is not parallel to Q because their slopes are not equal. ne pair of opposite sides is parallel, so QH is a trapezoid. et, use the distance formula to see whether an pairs of sides are congruent. Q ( ( )) ( ) " H (0 ) ( ) " H ( ( )) ( ) Q ( 0 ) ( ) 0 ecause Q H, QH is an isosceles trapezoid. Quick heck. a. Graph quadrilateral with vertices (, ), (, ), (, ), and (, ). b. lassif in as man was as possible. is a quadrilateral because it has four sides. It is a parallelogram because both pairs of opposite sides are parallel. It is a rhombus because all four sides are congruent. It is a rectangle because it has four right angles and its opposite sides are congruent. It is a square because it has four right angles and its sides are all congruent. ail otetaking Guide Geometr esson - ame lass ate Using lgebra Find the values of and in KM. K ( ) istribute. implif. he diagonals of a parallelogram ubstitute for in each other. the second equation to solve for. ubtract from each side. ivide each side b. ( ubstitute for in the first equation ) to solve for. o and. Quick heck. implif.. Find the value of in FGH. hen find m, m F, m G, and m H.. Find the values of a and b. M bisect ; ml 0, mlf 0, mlg 0, and mlh 0 a, b 0 0 ( ) X F G ( ) a b b 0 a H Y W Z Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

209 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - roving hat a Quadrilateral Is a arallelogram Is the Quadrilateral a arallelogram? an ou prove the quadrilateral is a parallelogram from what is given? plain. esson bjective etermine whether a quadrilateral is a parallelogram Ke oncepts. 00 trand: Geometr opic: Geometr heorem - If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. heorem - If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. heorem - If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. heorem - If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. amples. Finding Values for arallelograms Find values for and for which must be a parallelogram. If the diagonals of quadrilateral bisect each other, then is a parallelogram b heorem -. Write and solve two equations to find values of and for which the diagonals bisect each other. 0 iagonals of parallelograms bisect each other. ollect the variables on one side. 0 olve. If and, then is a parallelogram. Geometr esson - ail otetaking Guide 0 0 G 0 ame lass ate esson - esson bjectives Use properties of diagonals of rhombuses and rectangles etermine whether a parallelogram is a rhombus or a rectangle Ke oncepts. 00 trand: Geometr opic: Geometr hombuses heorem - ach diagonal of a rhombus bisects two angles of the rhombus. bisects, so bisects, so heorem -0 he diagonals of a rhombus are perpendicular. ' ectangles heorem - he diagonals of a rectangle are congruent. arallelograms heorem - If one diagonal of a parallelogram bisects two angles of the parallelogram, the parallelogram is a rhombus. heorem - If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. heorem - pecial arallelograms Given: m m G, m F 0 rove: FGH is a parallelogram. he sum of the measures of the angles of a polgon is F 0 (n )0 where n represents the number of sides, so the sum of the measures of the angles of a quadrilateral is ( )0 0. If represents the measure of the unmarked angle, 0 0, so = 0. heorem - states If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ecause both pairs of opposite angles are congruent, the quadrilateral is a parallelogram b heorem -. Quick heck.. Find the values of a and c for which Q must be a parallelogram. a 0, c. an ou prove the quadrilateral is a parallelogram? plain. Given: Q >, Q rove: Q is a parallelogram. Yes; a pair of opposite sides are parallel and congruent (heorem -). ail otetaking Guide Geometr esson - Q G H Q (a 0) a c c ame lass ate amples. Finding ngle Measures Find the measures of the numbered angles in the rhombus. heorem - states that each diagonal of a rhombus bisects two angles of the rhombus, so m. heorem -0 states that the diagonals of the rhombus are perpendicular, so m 0. ecause the four angles formed b the diagonals complementar all must have measure 0, and must be. ecause m, m 0. Finall, because, the Isosceles riangle heorem allows ou to conclude that. o m. Finding iagonal ength ne diagonal of a rectangle has length. he other diagonal has length. Find the length of each diagonal. heorem -, the diagonals of a rectangle are congruent. iagonals of a rectangle are congruent. ubtract from each side. ubtract from each side. ivide each side b. ( ) ubstitute. ( ) ubstitute. he length of each diagonal is. Quick heck.. Find the measures of the numbered angles in the rhombus. m 0, m 0, m 0, m 0. Find the length of the diagonals of rectangle GF if F and G =. F 0 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. G Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

210 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective Verif and use properties of trapezoids and kites 00 trand: Geometr opic: elationships mong Geometric Figures rapezoids and Kites Finding ngle Measures in Kites Find m, m, and m in the kite. Vocabular and Ke oncepts. rapezoids heorem - he base angles of an isosceles trapezoid are congruent. heorem - he diagonals of an isosceles trapezoid are congruent. he base angles of a trapezoid are two angles that share a base of the trapezoid. Kites heorem - he diagonals of a kite are perpendicular. ase angles eg ase angles amples. Finding ngle Measures in rapezoids WXYZ is an isosceles trapezoid, and m X. Find m Y, m Z, and m W. W wo angles that share m X m W 0 a leg of a trapezoid are supplementar. m W 0 ubstitute. m W ubtract from each side. eg X ecause the base angles of an isosceles trapezoid are congruent, m Y m X and m Z m W. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjective ame coordinates of special figures b using their properties ample. roving ongruenc how that WVU is a parallelogram b proving pairs of opposite sides congruent. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram b heorem -. You can prove that WVU is a parallelogram b showing that W VU and WV U. Use the distance formula. Use the coordinates (a, b), W( a c, b d), V( c e, d ), and U(e, 0). ( ) ( ) 00 trand: Geometr opic: osition and irection W a c a b d b c d VU ( c e e ) ( d 0 ) c d WV ( a c c e ) b d d ) U a e b 0 (a e) b ( ( ) ( ) ecause W VU and WV U, WVU is a parallelogram. Quick heck.. Use the diagram above. Use a different method: how that WVU is a parallelogram b finding the midpoints of the diagonals. Midpoint of V a a c e, midpoint of UW. hus, the b d b diagonals bisect each other, and WVU is a parallelogram. W(a c, b d) (a, b) V(c e, d) U(e, 0) Y Z lacing Figures in the oordinate lane (a e) b U perpendicular m 0 iagonals of a kite are. U efinition of a kite m Isosceles riangle heorem m m U 0 riangle ngle-um heorem m U 0 iagonals of a kite are perpendicular. 0 0 m ubstitute. 0 m implif. m ubtract from each side. Quick heck.. In the isosceles trapezoid, m 0. Find m, m Q, and m. 0, 0, 0. Find m, m, and m in the kite. m 0, m, and m ail otetaking Guide Geometr esson - 0 ame lass ate ample. aming oordinates Use the properties of parallelogram to find the missing coordinates. o not use an new variables. he verte is the origin with coordinates (0, 0). ecause point is p units to the left of point, point is also p units to the left of point, because is a parallelogram. o the first coordinate of point is p. ecause and is horizontal, is also horizontal. o point has the same second coordinate, q, as point. he missing coordinates are (0, 0) and ( p, q). Quick heck.. Use the properties of parallelogram Q to find the missing coordinates. o not use an new variables. Q(s b, c) (, 0) (s, 0) Q ( p, q) (b, c) Q(?,?) 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 Geometr ll-in-ne nswers Version

211 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective rove theorems using figures in the coordinate plane roofs Using oordinate Geometr 00 trand: Geometr opics: osition and irection; Mathematical easoning If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle from heorem. o show that XYZW is a rectangle, find the lengths of its diagonals, and then compare them to show that the are equal. Vocabular and Ke oncepts. heorem -: rapezoid Midsegment heorem () he midsegment of a trapezoid is parallel to the bases. () he length of the midsegment of a trapezoid is half the sum of the lengths of the bases. he midsegment of a trapezoid is the segment that joins the midpoints of the nonparallel opposite sides of the trapezoid. M, M, and M ( ) ample. Using oordinate Geometr Use coordinate geometr to prove that the quadrilateral formed b connecting the midpoints of rhombus is a rectangle. raw quadrilateral XYZW b connecting the midpoints of. Z( a, b) ( a, 0) Y( a, b) M (0, b) W(a, b) (a, 0) X(a, b) (0, b) From esson -, ou know that XYZW is a parallelogram. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjective Write ratios and solve proportions Vocabular and Ke oncepts. roperties of roportions a c b d is equivalent to () ad bc () b d () a b () a b a c c d b proportion is a statement that two ratios are equal. a c b d and a : b c : d are eamples of proportions. n etended proportion is a statement that three or more ratios are equal. is an eample of an etended proportion. he ross-roduct ropert states that the product of the etremes of a proportion is equal to the product of the means. scale drawing corresponding actual lengths. means a : b c : d etremes 00 trand: Geometr opic: osition and irection a d b c is a drawing in which all lengths are proportional to scale is the ratio of an length in a scale drawing to the corresponding actual length. he lengths ma be in different units. amples. Finding atios scale model of a car is in. long. he actual car is ft long. What is the ratio of the length of the model to the length of the car? Write both measurements in the same units. ft in. 0 in. length of model length of car in. ft 0 in. in. 0 a b c d atios and roportions c d d XZ ( a a ) (b ( b )) ( a ) ( b ) a b YW ( a a ) ( b ( b )) ( a ) ( b ) a b XZ YW ecause the diagonals are congruent, parallelogram XYZW is a rectangle. Quick heck. Use the diagram from the ample on page. plain wh the proof using (a, 0), (0, b), ( a, 0), and (0, b) is easier than the proof using (a, 0), (0, b), ( a, 0), and (0, b). Using multiples of in the coordinates for,,, and eliminates the use of fractions when finding midpoints, since finding midpoints requires division b. ail otetaking Guide Geometr esson - ame lass ate olving for a Variable olve each proportion. a. n n ( ) ross-roduct ropert n n 0 implif. ivide each side b. b. ( ) ross-roduct ropert istributive ropert ubtract from each side. Quick heck.. photo that is in. wide and in. high is enlarged to a poster that is ft wide and ft high. What is the ratio of the height of the photo to the height of the poster? :. olve each proportion. a. b. n n z 0 z 0. n he ratio of the length of the scale model to the length of the car is :. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

212 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - imilar olgons ample. esson bjectives Identif similar polgons ppl similar polgons 00 trand: Geometr and Measurement opics: ransformation of hapes and reservation of roperties; Measuring hsical ttributes Using imilar Figures If YXZ, find the value of. Y ecause YXZ, ou can write and solve a proportion. 0 0 orresponding sides are YZ X Z XZ proportional 0. Vocabular. imilar figures have the same shape but not necessaril the same size. wo polgons are similar if corresponding angles are congruent and corresponding sides are proportional. he mathematical smbol for similarit is. he similarit ratio is the ratio of the lengths of corresponding sides of similar figures. is a rectangle that can be divided into a square and a rectangle that is similar to the original rectangle. he golden ratio is the ratio of the length to the width of an golden rectangle, about. :. Quick heck. golden rectangle ample. Understanding imilarit XYZ. omplete each statement. a. m Y and m Y, so m because congruent angles have the same measure. b. YZ XZ corresponds to, so XZ YZ. XZ. efer to the diagram for ample. omplete: m and YZ XY Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Use,, and similarit statements ppl,, and similarit statements Vocabular and Ke oncepts. ostulate -: ngle-ngle imilarit ( ) ostulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. M heorem -: ide-ngle-ide imilarit ( ) heorem If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar. heorem -: ide-ide-ide imilarit ( ) heorem If the corresponding sides of two triangles are proportional, then the triangles are similar. Indirect measurement is a wa of measuring things that are difficult to measure directl. amples. Using the ostulate MX '. plain wh the triangles are similar. Write a similarit statement. ecause MX ', XM and XK are right angles, so XM > XK. > because their measures are equal. X Y 00 trand: Geometr opic: ransformation of hapes and reservation of roperties M Z roving riangles imilar M K Quick heck ubstitute. olve for. implif.. efer to the diagram for ample. Find. 0 ail otetaking Guide Geometr esson - ame lass ate Using imilarit heorems plain wh the triangles must be similar. Write a similarit statement. YVZ > WVX because the are vertical angles. VY VW and VZ VX, so corresponding sides are proportional. herefore, YVZ WVX b the ide-ngle-ide imilarit ( ) heorem. Quick heck.. In ample, ou have enough information to write a similarit statement. o ou have enough information to find the similarit ratio? plain. o; none of the side lengths are given.. plain wh the triangles at the right must be similar. Write a similarit statement. G GF F, so the triangles are similar b the ~ heorem; n, nfg Z V Y W G X F MX KX b the ngle-ngle imilarit ( ) ostulate. X Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

213 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective Find and use relationships in similar right triangles imilarit in ight riangles 00 trand: Geometr opic: ransformation of hapes and reservation of roperties Finding istance t a golf course, Maria drove her ball d straight toward the cup. Her brother Gabriel drove his ball straight 0 d, but not toward the cup. he diagram shows the results. Find and, their remaining distances from the cup. Use orollar of heorem - to solve for. 0 Write a proportion. 0 G up M 0 Vocabular and Ke oncepts. 0 heorem - he altitude to the hpotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and similar to each other. orollar to heorem - he length of the altitude to the hpotenuse of a right triangle is the geometric mean of the lengths of the segments of the hpotenuse. orollar to heorem - he altitude to the hpotenuse of a right triangle separates the hpotenuse so that the length of each leg of the triangle is the geometric mean of the length of the adjacent hpotenuse segment and the length of the hpotenuse. he geometric mean of two positive numbers a and b is the positive number such that a b. amples. Finding the Geometric Mean Find the geometric mean of and. Write a proportion.! ross-roduct ropert. Find the positive square root. he geometric mean of and is. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Use the ide-plitter heorem Use the riangle-ngle-isector heorem Ke oncepts. heorem -: ide-plitter heorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionall. heorem -: riangle-ngle-isector heorem If a ra bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. orollar to heorem - If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. amples. Using the ide-plitter heorem Find. M M 0 a b ide-pitter heorem ubstitute. 00 trand: Geometr opic: ransformation of hapes and reservation of roperties c d a b c d roportions in riangles M 0 ross-roduct ropert,,00 istributive ropert 0, olve for. 0 Use orollar of heorem - to solve for. ( ) ,00!,00 0 Write a proportion. ubstitute 0 for. implif. ross-roduct ropert Find the positive square root. Maria s ball is 0 d from the cup, and Gabriel s ball is 0 d from the cup. Quick heck.. Find the geometric mean of and 0.. ecall ample. Find the distance between Maria s ball and Gabriel s ball. 0 d ail otetaking Guide Geometr esson - ame lass ate Using the riangle-ngle-isector heorem Find the value of. G 0 I K 0 H Quick heck.. Use the ide-plitter heorem to find the value of.. 0. Find the value of.. IG GH 0 IK HK 0 ( 0 ) riangle-ngle-isector heorem. ubstitute. olve for ross-roduct ropert. olve for. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

214 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Use the thagorean heorem Use the onverse of the thagorean heorem he thagorean heorem and Its onverse 00 trand: Geometr opic: elationships mong Geometric Figures Is It a ight riangle? Is this triangle a right triangle? a b 0 c 0 ubstitute for a, for b, and for c. 0 implif. m m m Vocabular and Ke oncepts. ecause a b c, the triangle is not a right triangle. heorem -: thagorean heorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hpotenuse. a b c heorem -: onverse of the thagorean heorem If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. thagorean triple is a set of nonzero whole numbers a, b, and c that satisf the equation a b c. amples. thagorean riples Find the length of the hpotenuse of #. o the lengths of the sides of # form a thagorean triple? a b c Use the thagorean heorem. c ubstitute for a, and for b. c implif. Î c ake the square root. c implif. he length of the hpotenuse is. he lengths of the sides,,, and form a thagorean triple because the are whole numbers that satisf a b c. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Use the properties of - -0 triangles Use the properties of triangles Ke oncepts. heorem -: - -0 riangle heorem In a - -0 triangle, both legs are congruent and the length of the hpotenuse is " times the length of a leg. hpotenuse "? leg heorem -: riangle heorem In a triangle, the length of the hpotenuse is twice the length of the shorter leg. he length of the longer leg is " times the length of the shorter leg. hpotenuse? shorter leg longer leg "? shorter leg amples. Finding the ength of the Hpotenuse Find the value of the variable. Use the - -0 riangle heorem to find the hpotenuse. h "?! hpotenuse "? leg h " implif. h ( ) h ( ) h 0 " 00 trand: Geometr opic: elationships mong Geometric Figures h c 0 s b s s s a pecial ight riangles s 0 s Quick heck.. right triangle has a hpotenuse of length and a leg of length 0. Find the length of the other leg. o the lengths of the sides form a thagorean triple?!; no. triangle has sides of lengths,, and 0. Is the triangle a right triangle? no ail otetaking Guide Geometr esson - ame lass ate Using the ength of ne ide Find the value of each variable. Use the riangle heorem to find the lengths of the legs.!? hpotenuse? shorter leg 0 " ivide each side b. " implif. 0? " shorter leg riangle heorem "?! ubstitute " for shorter leg.? "? " implif. he length of the shorter leg is ", and the length of the longer leg is. Quick heck.. Find the length of the hpotenuse of a - -0 triangle with legs of length!.!. Find the lengths of the legs of a triangle with hpotenuse of length. ;! he length of the hpotenuse is 0". Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

215 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective Use tangent ratios to determine side lengths in triangles Vocabular. 00 trand: Measurement opic: Measuring hsical ttributes he tangent of acute in a right triangle is the ratio of the length of the leg opposite l to the length of the leg adjacent to l. he angent atio Using a angent atio o measure the height of a tree, lma walked ft from the tree and measured a angle from the ground to the top of the tree. stimate the height of the tree. ft he tree forms a right tangent ratio to estimate the height of the tree. angle with the ground, so ou can use the tangent of opposite You can abbreviate the equation as tan adjacent. amples. Writing angent atios Write the tangent ratios for and. 0 eg adjacent to opposite tan adjacent opposite tan adjacent eg length of leg opposite l length of leg adjacent to l 0 0 opposite Geometr esson - ail otetaking Guide ame lass ate esson - esson bjective Use sine and cosine to determine side lengths in triangles Vocabular. he sine of is the ratio of the length of the leg opposite to the length of the hpotenuse. he cosine of is the ratio of the length of the leg adjacent to to the hpotenuse. leg opposite l sine of hpotenuse hpotenuse eg opposite opposite his can be abbreviated sin hpotenuse. cosine of l leg adjacent to / eg hpotenuse adjacent to adjacent his can be abbreviated cos hpotenuse. amples. Writing ine and osine atios Use the triangle to find sin, cos, sin G, and cos G. Write our answers in simplest terms. opposite sin hpotenuse 0 adjacent cos hpotenuse 0 opposite sin G hpotenuse 0 adjacent cos G hpotenuse 0 00 trand: Measurement opic: Measuring hsical ttributes ine and osine atios 0 G tan Use the tangent ratio. height ( tan ) olve for height..0 Use a calculator. he tree is about feet tall. Quick heck. height. Write the tangent ratios for K and J.,. Find the value of w to the nearest tenth. a. 0 b. w. w.0 ail otetaking Guide Geometr esson - ame lass ate Using the osine atio 0-ft wire supporting a flagpole forms a angle with the flagpole. o the nearest foot, how high is the flagpole? he flagpole, wire, and ground form a right triangle with the wire as the hpotenuse. ecause ou know an angle and the measure of its hpotenuse, ou can use the cosine ratio to find the height of the flagpole. height cos Use the cosine ratio. 0 height 0? cos olve for height. 0.0 Use a calculator. he flagpole is about feet tall. Quick heck.. Write the sine and cosine ratios for X and Y. sin X, cos X, sin Y, cos Y In ample, suppose that the angle the wire makes with the ground is 0. What is the height of the flagpole to the nearest foot? feet. J Z 0 ft X 0 0 ft 0 K Y 00 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 ll-in-ne nswers Version Geometr

216 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective 00 trand: Measurement ngles of levation and epression viation n airplane fling 00 ft above the ground begins a descent to land at the airport. How man miles from the airport is the airplane when it starts its descent? (ote: he angle is not drawn to scale.) Use angles of elevation and depression to solve problems opic: Measuring hsical ttributes,00 ft ` Vocabular. Ground 0 n angle of elevation is the angle formed b a horizontal line and the line of sight to an object above the horizontal line. n sight to an object below the horizontal line. amples. angle of depression is the angle formed b a horizontal line and the line of Identifing ngles of levation and epression escribe and as the relate to the situation shown. ngle of depression Horizontal line Horizontal line ngle of elevation ne side of the angle of depression is a horizontal line. is the angle of depression from the airplane to the building. ne side of the angle of elevation is a horizontal line. is the angle of elevation from the building to the airplane. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives escribe vectors olve problems that involve vector addition Vocabular and Ke oncepts. dding Vectors For a, and c,, a c k, l. vector is an quantit with magnitude (size) and direction. vector can be represented with an arrow. he magnitude of a vector is its size, or length. he initial point of a vector is the point at which it starts. he terminal point of a vector is the point at which it ends. resultant vector is the sum of other vectors. 00 trand: Geometr opic: osition and irection ecause point M is in the third quadrant, both coordinates are negative.o the nearest tenth, M k.,.l. Vectors amples. escribing a Vector escribe M as an ordered pair. Give coordinates to the nearest tenth. 0 Use the sine and cosine ratios to find the values of and. 0 cos 0 Use sine and cosine. sin 0 M ( cos 0 ) olve for the variable. 0( sin 0 ). Use a calculator he airplane is about Quick heck. sin,00 00 sin Use the olve for. Use a calculator. ivide b 0 feet to miles. miles from the airport when it starts its descent.. escribe each angle as it relates to the situation in ample. a. he angle of depression from the building to the person on the ground b. he angle of elevation from the person on the ground to the building. n airplane pilot sees a life raft at a angle of depression. he airplane s altitude is km. What is the airplane s surface distance d from the raft? about. km ail otetaking Guide Geometr esson - sine ratio. to convert ame lass ate dding Vectors Vectors v, and w, are shown at the right. Write the sum of the two vectors as an ordered pair. hen draw s, the sum of v and w. o find the first coordinate of s, add the first coordinates of v and w. o find the second coordinate of s, add the second coordinates of v and w. s k, l k, l k, ( ) l dd the coordinates. k, 0 l implif. raw vector v using the origin as the initial point. raw vector w using the terminal point of v, (, ), as the initial point. raw the resultant vector s from the initial point of v to the terminal point of w. Quick heck.. escribe the vector at the right as an ordered pair. Give the coordinates to the nearest tenth. k.,.l. Write the sum of the two vectors k, l and k, l as an ordered pair. k, l 0 (, ) (, ) v w s 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 Geometr ll-in-ne nswers Version

217 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - ranslations Finding a ranslation Image Find the image of under the translation (, ) (, ). esson bjectives Identif isometries Find translation images of figures Vocabular. 00 trand: Geometr opic: ransformation of hapes and reservation of roperties; osition and irection (, ) (, ) Use the rule (, ) (, ) (, 0) (, 0 ) (0, ) ( 0, ) he image of is with (, ), (, ), and (, ). transformation of a geometric figure is a change in its position, shape, or size. Quick heck. 0 In a transformation, the are made. is the original image before changes In a transformation, the image is the resulting figure after changes are made. n are congruent. is a transformation in which the preimage and the image translation (slide) is a transformation that maps all points the same distance and in the same direction. composition of transformations amples. Identifing Isometries oes the transformation appear to be an isometr? Image reimage isometr preimage is a combination of two or more transformations. he image appears to be the same as the preimage, but turned. ecause the figures appear to be congruent, the transformation appears to be an isometr. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Find reflection images of figures Vocabular. 00 trand: Geometr opics: ransformation of hapes and reservation of roperties eflections reflection in line r is a transformation such that if a point is on line r, then the image of is itself, and if a point is not on line r, then its image is the point such that r is the perpendicular bisector of r. r ine of reflection ample. Finding eflection Images If point Q(, ) is reflected across line, what are the coordinates of its reflection image? Q is units to the left of the reflection line, so its image Q is units to the right of the reflection line. he reflection line is the perpendicular bisector of QQr if Q is at (, ). Quick heck.. What are the coordinates of the image of Q if the reflection line is? (, ) Q =. oes the transformation appear to be an isometr? plain. a. b. reimage reimage Image. Find the image of for the translation (, ). (, ), (, ), (, 0) Yes; the figures appear to be congruent b a flip. Image Yes; the figures appear to be congruent b a flip and a slide. ail otetaking Guide Geometr esson - ame lass ate amples. rawing eflection Images XYZ has vertices X(0, ), Y(, 0), and Z(, ). raw XYZ and its reflection image in the line. First locate vertices X, Y, and Z and draw XYZ in a coordinate plane. ocate points X, Y, and Z such that the line of reflection is the perpendicular bisector of XXr, YYr, and ZZr. raw the reflection image X Y Z. ongruent ngles is the reflection of across. how that and W form congruent angles with line. ecause a reflection is an isometr, and r form congruent angles with line. r and W also form congruent angles with line because vertical angles are congruent. herefore, and W form congruent angles with line b the ransitive ropert. Quick heck.. has vertices (, ), (0, ), and (, ). raw and its reflection image in the line.. In ample, what kind of triangle is? Imagine the image of point W reflected across line. What can ou sa about WW and? isosceles; the are similar b W 0 X ZZ X Y Y 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - 0 ll-in-ne nswers Version Geometr

218 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective raw and identif rotation images of figures 00 trand: Geometr opic: ransformation of hapes and reservation of roperties otations Identifing a otation Image egular heagon F is divided into si equilateral triangles. a. ame the image of for a 0 rotation about M. ecause 0 0, each central angle of F measures 0. 0 counterclockwise rotation about center M moves point across four triangles. he image of point is point. F M Vocabular. rotation of about a point is a transformation for which the following are true: he image of is itself (that is, ). For an point V, V V and m VV. amples. rawing a otation Image raw the image of under a 0 rotation about. tep Use a protractor to draw a 0 angle at verte with one side. tep Use a compass to construct r. tep ocate and in a similar manner. hen draw. 0 0 Geometr esson - ail otetaking Guide V V ame lass ate esson - esson bjective Identif the tpe of smmetr in a figure Vocabular. 00 trand: Geometr opic: ransformation of hapes and reservation of roperties figure has smmetr if there is an isometr that maps the figure onto itself. figure has reflectional smmetr if there is smmetr that maps the figure onto itself. ine smmetr is the same as reflectional smmetr. ine of mmetr he heart-shaped figure has line ( ) smmetr. reflectional figure has rotational smmetr if it is its own image for some rotation of 0 or less. figure has point smmetr if it has 0 rotational smmetr. he figure is its own image after one half-turn, so it has rotational smmetr with a 0 angle of rotation. he figure also has point smmetr. mmetr b. ame the image of M for a 0 rotation about F. MF is equilateral, so FM has measure rotation of MF about point F would superimpose FM on F, so the image of M under a 0 rotation about point F is point. Quick heck.. raw the image of for a 0 rotation about point. abel the vertices of the image.. egular pentagon is divided into congruent triangles. ame the image of for a rotation about point X. ail otetaking Guide Geometr esson - ame lass ate amples. Identifing ines of mmetr raw all lines of smmetr for the isosceles trapezoid. here is one raw an lines that divide the isosceles trapezoid so that half of the figure is a mirror image of the other half. line of smmetr. Identifing otational mmetr Judging from appearance, do the letters V and H have rotational smmetr? If so, give an angle of rotation. he letter V does not have rotational smmetr because it must be rotated 0 before it is its own image. he letter H is its own image after one half-turn, so it has rotational smmetr with a 0 angle of rotation. Quick heck.. raw a rectangle and all of its lines of smmetr.. a. Judging from appearance, tell whether the figure at the right has rotational smmetr. If so, give the angle of rotation. es; 0 X b. oes the figure have point smmetr? es Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

219 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective ocate dilation images of figures ilations 00 trand: Geometr opic: Geometr cale rawings he scale factor on a museum s floor plan is : 00. he length and width of one wing on the drawing are in. and in. Find the actual dimensions of the wing in feet and inches. he floor plan is a reduction of the actual dimensions b a scale factor of 00. Multipl each dimension on the drawing b 00 to find the actual dimensions. hen write the dimensions in feet and inches. Vocabular. dilation is a transformation with center and scale factor n for which the following are true: he image of is itself ) (that is, ). For an point, is on and n. Q rqr is the image of Q under a with center and scale factor. dilation Q n enlargement is a dilation with a scale factor greater than. reduction is a dilation with a scale factor less than. amples. Finding a cale Factor ircle with -cm diameter and center is a dilation of concentric circle with -cm diameter. escribe the dilation. he circles are concentric, so the dilation has center. ecause the diameter of the dilation image is smaller, the dilation is a reduction. diameter of dilation image cale factor: diameter of preimage he dilation is a reduction with center and scale factor. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Use a composition of reflections Identif glide reflections Vocabular and Ke oncepts. heorem - translation or rotation is a composition of two reflections. heorem - composition of reflections across two parallel lines is a translation. heorem - composition of reflections across two intersecting lines is a rotation. heorem -: Fundamental heorem of Isometries In a plane, one of two congruent figures can be mapped onto the other b a composition of at most three reflections. heorem -: Isometr lassification heorem here are onl four isometries. he are the following: ompositions of eflections 00 trand: Geometr opic: ransformation of hapes and reservation of roperties eflection ranslation otation Glide reflection glide reflection is the composition of a glide (translation) and a reflection across a line parallel to the direction of translation. 00 in. 00 in. ft, in. 00 in. 00 in. 00 ft ft, in. 00 ft he museum wing measures b. Quick heck.. Quadrilateral J K M is a dilation image of quadrilateral JKM. escribe the dilation. J J K K M M he dilation is a reduction with center (0, 0) and scale factor.. he height of a tractor-trailer truck is. m. he scale factor for a model truck is. Find the height of the model to the nearest centimeter. cm ail otetaking Guide Geometr esson - ame lass ate ample. omposition of eflections in Intersecting ines he letter is reflected in line and then in line. escribe the resulting rotation. Find the image of through a reflection across line. Find the image of the reflection through another reflection across line. he composition of two reflections across intersecting lines is a rotation. he center of rotation is the point where the lines intersect, and the angle is twice the angle formed b the intersecting lines. o, the letter is rotated clockwise, or counterclockwise, with the center of rotation at point. Quick heck. a. eflect the letter across a and then b. escribe the resulting rotation. nswers ma var. he result is a clockwise rotation about point c through an angle of mlacb. b. Use parallel lines and m. raw between and m. Find the image of for a reflection across line and then across line M. escribe the resulting translation. nswers ma var. he result is a translation twice the distance between and m. c m a b Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

220 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Identif transformation in tessellations, and figures that will tessellate Identif smmetries in tessellations essellations 00 trand: Geometr opic: Geometr Identifing mmetries in essellations ist the smmetries in the tessellation. tarting at an verte, the tessellation can be mapped onto itself using a 0 rotation, so the tessellation has point smmetr centered at an verte. Vocabular and Ke oncepts. heorem - ver triangle tessellates. heorem - ver quadrilateral tessellates. tessellation, or tiling, is a repeating pattern of figures that completel covers a plane, without gaps or overlaps. ranslational smmetr is the tpe of smmetr for which there is a translation that maps a figure onto itself. Glide reflectional smmetr is the tpe of smmetr for which there is a glide reflection that maps a figure onto itself. amples. Identifing the ransformation in a essellation Identif the repeating figures and a transformation in the tessellation. repeated combination of an octagon and one adjoining square will completel cover the plane without gaps or overlaps. Use arrows to show a translation. Geometr esson - ail otetaking Guide ame lass ate esson 0- esson bjectives Find the area of a parallelogram Find the area of a triangle Vocabular and Ke oncepts. heorem 0-: rea of a ectangle he area of a rectangle is the product of its base and height. bh heorem 0-: rea of a arallelogram he area of a parallelogram is the product of a base and the corresponding height. bh heorem 0-: rea of a riangle he area of a triangle is half the product of a base and the corresponding height. bh 00 trand: Measurement opic: Measuring hsical ttributes base of a parallelogram is an of its sides. he altitude of a parallelogram corresponding to a given base is the segment perpendicular to the line containing that base drawn from the side opposite the base. he height of a parallelogram is the length of its altitude. base of a triangle is an of its sides. he height of a triangle is the length of the altitude to the line containing that base. b reas of arallelograms and riangles h h b h b ltitude ase h he tessellation also has translational smmetr, as can be seen b sliding an triangle onto a cop of itself along an of the lines. Quick heck.. Identif a transformation and outline the smallest repeating figure in the tessellation below. translation. ist the smmetries in the tessellation. line smmetr, rotational smmetr, glide reflectional smmetr, translational smmetr ail otetaking Guide Geometr esson - ame lass ate amples. Finding the rea of a arallelogram parallelogram has -in. and -in. sides. he height corresponding to the -in. base is in. Find the area of the parallelogram. bh rea of a parallelogram ( ) ubstitute for b and for h. implif. he area of the parallelogram is in. Finding the rea of a riangle Find the area of XYZ. bh rea of a triangle X ( 0 )( ) ubstitute 0 for b and for h. cm cm implif. XYZ has area cm. Y 0 cm Quick heck.. parallelogram has sides cm and cm. he height corresponding to a -cm base is cm. Find the area of the parallelogram. cm. Find the area of the triangle. cm cm 0 cm cm Z b 0 Geometr esson 0- ail otetaking Guide ail otetaking Guide Geometr esson 0-0 Geometr ll-in-ne nswers Version

221 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson 0- esson bjectives Find the area of a trapezoid Find the area of a rhombus or a kite Vocabular and Ke oncepts. heorem 0-: rea of a rapezoid reas of rapezoids, hombuses, and Kites 00 trand: Measurement opic: Measuring hsical ttributes Finding the rea of a hombus Find the area of rhombus U. o find the area, ou need to know the lengths of both diagonals. raw diagonal U, and label the intersection of the diagonals point X. X is a right triangle because the diagonals of a rhombus are perpendicular. he diagonals of a rhombus bisect each other, so X ft. You can use the thagorean triple,, or the thagorean heorem to conclude that X ft. U 0 ft because the diagonals of a rhombus bisect each other. U ft X ft he area of a trapezoid is half the product of the height and the sum of the bases. heorem 0-: rea of a hombus or a Kite he area of a rhombus or a kite is half the product of the lengths of its diagonals. he height of a trapezoid is the perpendicular distance h between the bases. amples. ppling the rea of a rapezoid car window is shaped like the trapezoid shown. Find the area of the window. h(b b ) d d h( b b ) rea of a trapezoid ( )( 0 ) ubstitute for h, 0 for b, and for b. 0 implif. he area of the car window is 0 in.. in. b h 0 in. Geometr esson 0- ail otetaking Guide Vocabular and Ke oncepts. heorem 0-: rea of a egular olgon he area of a regular polgon is half the product of the apothem and the perimeter. ap eg ase d b h b ase ame lass ate esson 0- esson bjective Find the area of a regular polgon he center of a regular polgon is the center of the circumscribed circle. he radius of a regular polgon is the distance from the center to a verte. he apothem of a regular polgon is the perpendicular distance from the center to a side. amples. Finding ngle Measures his regular heagon has an apothem and radii drawn. Find the measure of each numbered angle. m 0 0 ivide 0 b the number of sides. he apothem bisects the verte angle of m m the isosceles triangle formed b the radii. m ( 0 ) 0 ubstitute 0 for m. b d eg in. reas of egular olgons 00 trand: Measurement opic: Measuring hsical ttributes enter adius pothem a p d d rea of a rhombus implif. he area of rhombus U is 0 ft. ( )( ) ubstitute for d and for d. Quick heck.. Find the area of a trapezoid with height cm and bases cm and cm.. cm. ritical hinking In ample, eplain how ou can use a thagorean triple to conclude that XU ft. ail otetaking Guide Geometr esson 0- ame lass ate.finding the rea of a egular olgon librar is in the shape.0 ft of a regular octagon. ach side is.0 ft. he radius of the octagon is. ft. Find the area of the librar to the nearest 0 ft. onsecutive radii form an isosceles triangle, so an apothem bisects the side of the octagon.. ft o appl the area formula ap, ou need to find a and p. tep Find the apothem a..0 ft.0 ft a (.0 ) (. ) thagorean heorem. a olve for a. a a <.. tep Find the perimeter p. p ns Find the perimeter, where n the number of sides of a regular polgon..0 p ()(.0 ) ubstitute for n and for s, and simplif. tep Find the area. ap rea of a regular polgon.. ( )( ) ubstitute for a and for p.. implif. o the nearest 0 ft, the area is 0 ft. Quick heck.. t the right, a portion of a regular octagon has radii and an apothem drawn. Find the measure of each numbered angle. ml ; ml.; ml.. Find the area of a regular pentagon with.-cm sides and an -cm apothem. cm m 0 ( 0 0 ) 0 he sum of the measures of the angles of a triangle is 0. m 0, m 0, and m 0 Geometr esson 0- ail otetaking Guide ail otetaking Guide Geometr esson 0- ll-in-ne nswers Version Geometr

222 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson 0- esson bjective Find the perimeters and areas of similar figures erimeters and reas of imilar Figures 00 trand: Measurement and umber roperties and perations opics: stems of Measurement; atios and roportional easoning Using imilarit atios enita plants the same crop in two rectangular fields. ach dimension of the larger field is times the dimension of the smaller field. eeding the smaller field costs $. How much mone does seeding the larger field cost? he similarit ratio of the fields is. :, so the ratio of the areas of the fields is (.) : (), or. to. Ke oncepts. ecause seeding the smaller field costs $, seeding. times as much land costs. $. heorem 0-: erimeters and reas of imilar Figures If the similarit ratio of two similar figures is a b, then a () the ratio of their perimeters is b and a () the ratio of their areas is. b amples. Finding atios in imilar Figures he triangles at the right are similar.find the ratio (larger to smaller) of their perimeters and of their areas. he shortest side of the left-hand triangle has length, and the shortest side of the right-hand triangle has length. From larger to smaller, the similarit ratio is. the erimeters and reas of imilar Figures heorem, the ratio of the perimeters is, and the ratio of the areas is, or. Finding reas Using imilar Figures he ratio of the length of the corresponding sides of two regular octagons is. he area of the larger octagon is 0 ft. Find the area of the smaller octagon. ll regular octagons are similar. ecause the ratio of the lengths of the corresponding sides of the regular octagons is, the ratio of their areas is, or. Write a proportion. 0 0 Use the ross-roduct ropert. ivide each side b. he area of the smaller octagon is ft. Geometr esson 0- ail otetaking Guide ame lass ate esson 0- esson bjectives Find the area of a regular polgon using trigonometr Find the area of a triangle using trigonometr Ke oncepts. heorem 0-: rea of a riangle Given he area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. rea of bc(sin ) 00 trand: Measurement opic: Measuring hsical ttributes.. rigonometr and rea amples. Finding rea he radius of a garden in the shape of a regular pentagon is feet. Find the area of the garden. Find the perimeter p and apothem a, and then find the area using the formula ap. ft 0 ecause a pentagon has five sides, m. M and are radii, so. herefore, M M b the H heorem, so m M m. Use the cosine ratio to find a. Use the sine ratio to find M. cos a Use the ratio. sin M a (cos ) olve. M (sin ) Use M to find p. ecause M M,? M. ecause the pentagon is regular, p?. c b a eeding the larger field costs $. Quick heck.. wo similar polgons have corresponding sides in the ratio :. a. Find the ratio of their perimeters. b. Find the ratio of their areas. : :. he corresponding sides of two similar parallelograms are in the ratio. he area of the smaller parallelogram is in.. Find the area of the larger parallelogram. in.. he similarit ratio of the dimensions of two similar pieces of window glass is :. he smaller piece costs $.0. What should be the cost of the larger piece? $. ail otetaking Guide Geometr esson 0- ame lass ate (cos ) 0(sin ) ubstitute for a and p. 0 (cos ) (sin ) implif. 0. Use a calculator. he area is about 0 ft. urveing triangular park has two sides that measure 00 ft and 00 ft and form a angle. Find the area of the park to the nearest hundred square feet. Use heorem 0-: he area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. 00 ft rea side length side length sine of included angle heorem 0- rea sin ubstitute. rea 0,000 sin ubstitute.. Use a calculator. he area of the park is approimatel,00 ft. Quick heck.. Find the area of a regular octagon with a perimeter of 0 in. Give the area to the nearest tenth.. in.. wo sides of a triangular building plot are 0 ft and ft long. he include an angle of. Find the area of the building plot to the nearest square foot. 0 ft 00 ft o p (? M ) 0? M 0? (sin ) 0(sin ). Finall, substitute into the area formula ap. Geometr esson 0- ail otetaking Guide ail otetaking Guide Geometr esson 0- Geometr ll-in-ne nswers Version

223 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson 0- esson bjectives Find the measures of central angles and arcs Find circumference and arc length Vocabular and Ke oncepts trand: Measurement opic: Measuring hsical ttributes ostulate 0-: rc ddition ostulate he measure of the arc formed b two adjacent arcs is the sum of the measures of the two arcs. 0 0 m m m heorem 0-: ircumference of a ircle he circumference of a circle is p times the diameter. pd or pr heorem 0-0: rc ength he length of an arc of a circle is the product of the ratio measure of the arc and the circumference of the circle. 0 length of 0 m 0 0? pr circle is the set of all points equidistant from a given point called the center. d r r center of a circle is the point from which all points are equidistant. radius is a segment that has one endpoint at the center and the other endpoint on the circle. central angle is an angle whose verte is the center of the circle. ircumference of a circle is the distance around the circle. i ( ) is the ratio of the circumference of a circle to its diameter. ircles and rcs Geometr esson 0- ail otetaking Guide ame lass ate esson 0- esson bjective Find the areas of circles, sectors, and segments of circles Vocabular and Ke oncepts. heorem 0-: rea of a ircle he area of a circle is the product of p and the square of the radius. pr heorem 0-: rea of a ector of a ircle measure of the arc he area of a sector of a circle is the product of the ratio 0 and the area of the circle.. area of archer target p rea of sector sector of a circle is a region bounded b two radii and their intercepted arc. segment of a circle is the part bounded b an arc and the segment joining its endpoints. amples. ppling the rea of a ircle circular archer target has a -ft diameter. It is ellow ecept for a red bull s-ee at the center with a -in. diameter. Find the area of the ellow region to the nearest whole number. First find the areas of the archer target and the red bull s-ee. he radius of the archer target is ( ft) ft in. he area of the archer target is πr π( ) p in. he radius of the red bull s-ee region is ( in.) in. he area of the red region is πr π( ) p in. area of red region p 00 trand: Measurement opic: Measuring hsical ttributes 0 m 0? pr area of ellow region π r reas of ircles and ectors r egment of a circle ector of a circle amples. 0 Finding the Measures of rcs Find mxy and mxm in circle mxy mx my rc ddition ostulate 0 0 he measure of a minor arc is the measure mxy m/ X my of its corresponding central angle. 0 mxy 0 ubstitute. Y 0 mxy implif. 0 0 mxm mx mxwm rc ddition ostulate mxm 0 ubstitute. mxm implif. Finding rc ength Find the length of in circle M in terms of π. 0 ecause 0, m rc ddition ostulate length of m 0? pr rc ength ostulate 0 0 π( ) ubstitute. π he length of is π cm. Quick heck.. Use the diagram in ample. Find m Y, mymw, mmw 0, and mxmy. 0; 0; ;. Find the length of a semicircle with radius. m in terms of π..p m ail otetaking Guide Geometr esson 0- M X W 0 cm M ame lass ate Finding the rea of a ector of a ircle Find the area of sector. eave our answer in terms of π. m area of sector πr 0 π 0 π he area of sector is 0π m. π ( ) Quick heck.. How much more pizza is in a -in.-diameter pizza than in a -in. pizza? about in.. ritical hinking circle has a diameter of 0 cm. What is the area of a sector bounded b a 0 major arc? ound our answer to the nearest tenth.. cm 00 0 ubstitute. implif. implif. m 00.0 he area of the ellow region is about in. implif. Geometr esson 0- ail otetaking Guide ail otetaking Guide Geometr esson 0- ll-in-ne nswers Version Geometr

224 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson 0- esson bjective Use segment and area models to find the probabilities of events Geometric robabilit 00 trand: ata nalsis and robabilit opic: robabilit ample. Finding robabilit Using rea circle is inscribed in a square target with 0-cm sides. Find the probabilit that a dart landing randoml within the square does not land within the circle. Find the area of the square. 0 cm Vocabular. Geometric probabilit is a model in which ou let points represent outcomes. s 0 00 cm Find the area of the circle. ecause the square has sides of length 0 cm, the circle s diameter is 0 cm, so its radius is 0 cm. πr π( 0 ) 00π cm ample. Finding robabilit Using egments gnat lands at random on the edge of the ruler below. Find the probabilit that the gnat lands on a point between and 0. he length of the segment between and 0 is 0. he length of the ruler is. (landing between and 0) length of favorable segment length of entire segment Quick heck.. point on is selected at random. What is the probabilit that it is a point on? Geometr esson 0- ail otetaking Guide ame lass ate esson - esson bjective ecognize polhedra and their parts Visualize cross sections of space figures Vocabular and Ke oncepts. 00 trand: Geometr opic: imension and hape uler s Formula he numbers of faces (F), vertices (V), and edges () of a polhedron are related b the formula F V. polhedron is a three-dimensional figure whose surfaces are polgons. face is a flat surface of a polhedron in the shape of a polgon. n edge is a segment that is formed b the intersection of two faces. verte is a point where three or more edges intersect. cross section is the intersection of a solid and a plane. amples. Identifing Vertices, dges, and Faces How man vertices, edges, and faces are there in the polhedron shown? Give a list of each. here are four vertices:,,, and. here are si edges:,,,,, and. pace Figures and ross ections ectangular Faces dge Verte prism Find the area of the region between the square and the circle. (00 00π ) cm Use areas to calculate the probabilit that a dart landing randoml in the square does not land within the circle. Use a calculator. ound to the nearest thousandth. area between square and circle (between square and circle) area of square 00 00π π he probabilit that a dart landing randoml in the square does not land within the circle is about. %. Quick heck.. Use the diagram in ample. If ou change the radius of the circle as indicated, what then is the probabilit of hitting outside the circle? a. ivide the radius b. about 0.% b. ivide the radius b. about.% ail otetaking Guide Geometr esson 0- ame lass ate Using uler s Formula Use uler s Formula to find the number of edges on a solid with faces and vertices. F V uler s Formula ubstitute the number of faces and vertices. implif. solid with faces and vertices has edges. Quick heck.. ist the vertices, edges, and faces of the polhedron.,,, U, V;, U,, V, VU, V, U, U, ; ku, ku, k, kvu, kvu, kv. Use uler s Formula to find the number of edges on a polhedron with eight triangular faces. edges U V here are four faces: k, k, k, and k. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

225 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Find the surface area of a prism Find the surface area of a clinder Vocabular and Ke oncepts. urface reas of risms and linders 00 trand: Measurement opic: Measuring hsical ttributes clinder is a three-dimensional figure with two congruent circular bases that lie in parallel planes. In a clinder, the bases are parallel circles. he lateral area of a clinder is the area of the curved surface. ases right h h clinders heorem -: ateral and urface rea of a linder he lateral area of a right clinder is the product of the circumference of the base and the height of the clinder... πrh, or.. πdh he surface area of a right clinder is the sum of the lateral area and the areas of the two circular bases....., or.. πrh πr h πr ateral rea prism is a polhedron with eactl two congruent, parallel faces. he faces. bases r h urface rea of a polhedron are the parallel ateral faces are the faces on a prism that are not bases. he lateral area of a prism is the sum of the areas of the lateral faces. he surface area of a prism or a clinder is the sum of the lateral area and the areas of the two bases. Geometr esson - ail otetaking Guide Vocabular and Ke oncepts. ateral faces r r ases heorem -: ateral and urface rea of a egular ramid he lateral area of a regular pramid is half the product of the perimeter of the base and the slant height... p he surface area of a regular pramid is the sum of the lateral area and the area of the base..... entagonal riangular rea of a base π r prism ateral edges ases regular pramid is a pramid whose base is a regular polgon and whose lateral faces are congruent isosceles triangles. he altitude of a pramid or cone is the perpendicular segment from the verte to the plane of the base. ateral Verte he height of a pramid or a cone is the length of edge ateral face the altitude. ltitude he slant height of a regular pramid ase is the length of the altitude of a lateral face. he lateral area of a pramid is is the sum of the ase egular ramid edge areas of the congruent lateral faces. he surface area of a pramid is the sum of the lateral area and the area of thebase. ample. Finding urface rea of a ramid Find the surface area of a square pramid with base edges. ft and slant height ft. ft prism ame lass ate esson - esson bjectives Find the surface area of a pramid Find the surface area of a cone 00 trand: Measurement opic: Measuring hsical ttributes urface reas of ramids and ones ample. Finding urface rea of linders compan sells cornmeal and barle in clindrical containers. he diameter of the base of the -in.-high cornmeal container is in. he diameter of the base of the -in.-high barle container is in. Which container has the greater surface area? Find the surface area of each container. emember that r d. ornmeal ontainer arle ontainer.... Use the formula for surface area..... ubstitute the formulas for lateral prh p pr r p r h area of a clinder and area of a circle. p p implif. p p p ombine like terms. p π( )( ) π( ) ubstitute for r and h. π( )( ) π( ) ecause π in. π in., the barle container has the greater surface area. Quick heck. a. Find the surface area of a clinder with height 0 cm and radius 0 cm in terms of π. 00p cm b. he compan in the ample wants to make a label to cover the cornmeal container. he label will cover the container all the wa around, but will not cover an part of the top or bottom. What is the area of the label to the nearest tenth of a square inch?. in. ail otetaking Guide Geometr esson - ame lass ate he perimeter p of the square base is. ft, or 0 ft. You are given ft and ou found that p 0 ft, so ou can find the lateral area ( 0 )( ) ubstitute. 0 Use the formula for lateral area of a pramid. implif. Find the area of the square base. ecause the base is a square with side length. ft, s p Use the formula for surface area of a pramid. ubstitute. implif. he surface area of the square pramid is. ft. Quick heck. Find the surface area of a square pramid with base edges m and slant height m. m. ft 0 Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr

226 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjectives Find the volume of a prism Find the volume of a clinder Volumes of risms and linders 00 trand: Measurement opic: Measuring hsical ttributes Finding Volume of a riangular rism Find the volume of the prism. he prism is a right triangular prism with triangular bases. he base of the triangular prism is a right triangle where one leg is the base and the other leg is the altitude. Use the thagorean heorem to calculate the length of the other leg. 0 m m 0 m Vocabular and Ke oncepts. heorem -: avalieri s rinciple If two space figures have the same height and the same cross sectional area at ever level, then the have the same volume. heorem -: Volume of a rism he volume of a prism is the product of the area of a base and the height of the prism. V h heorem -: Volume of a linder he volume of a clinder is the product of the and the height of the clinder. V h, or V pr h area of a base Volume is the space that a figure occupies. composite space figure is a three-dimensional figure that is the combination of two or more simpler figures. amples. Finding Volume of a linder Find the volume of the clinder. eave our answer in terms of π. he formula for the volume of a clinder is V pr h. he diagram shows h and d, but ou must find r. r d V p p? r h? p Use the formula for the volume of a clinder. ubstitute. implif. he volume of the clinder is π ft. Geometr esson - ail otetaking Guide ame lass ate esson - esson bjectives Find the volume of a pramid Find the volume of the cone Ke oncepts. heorem -: Volume of a ramid he volume of a pramid is one third the product of the area of the base and the height of the pramid. h V h heorem -: Volume of a one he volume of a cone is one third the product of the area of the base and the height of the cone. V h h, or V pr h r amples. Finding Volume of a ramid Find the volume of a square pramid with base edges cm and height cm. ecause the base is a square,?. V h Use the formula for volume of a pramid. V 0 implif. he volume of the square pramid is 0 cm. 00 trand: Measurement opic: Measuring hsical ttributes ( )( ) ubstitute for and for h. h ft r h ft Volumes of ramids and ones area of the base to find the volume of the prism. V h Use the formula for the volume of a prism. V 0? 0 ubstitute. V 00 implif. he volume of the triangular prism is 00 m. he area of the base is bh ( 0 )( ) 0. Use the Quick heck.. he clinder shown is oblique. a. Find its volume in terms of π. p m b. Find its volume to the nearest tenth of a cubic meter. 0. m. Find the volume of the triangular prism. 0 m m 0 m ail otetaking Guide Geometr esson ame lass ate Finding Volume of an blique one Find the volume of the oblique cone in terms of π. r V p r d in. h Use the formula for the volume of a cone. ( )( ) ubstitute for r and for h. p p implif. he volume of the cone is p in.. Quick heck.. Find the volume of a square pramid with base edges m and slant height m. 0 m. small child s teepee is in the shape of a cone ft tall and ft in diameter. Find the volume of the teepee to the nearest cubic foot. ft m m in. in. m Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - Geometr ll-in-ne nswers Version

227 Geometr: ll-in-ne nswers Version (continued) ame lass ate ame lass ate esson - esson bjective Find the surface area and volume of a sphere Vocabular and Ke oncepts. heorem -0: urface rea of a phere he surface area of a sphere is four times the product of the square of the radius of the sphere... pr heorem -: Volume of a phere he volume of a sphere is four thirds the product of p and the cube of the radius of the sphere... pr p and sphere is the set of all points in space equidistant from a given point. he center of a sphere is the given point from which all points on the sphere are equidistant. he radius of a sphere is a segment that has one endpoint at the center and the other endpoint on the sphere. he diameter of a sphere is a segment passing through the center with endpoints on the sphere. great circle is the intersection of a sphere and a plane containing the center of the sphere. he great circle divides a sphere into two congruent hemispheres. he circumference of a sphere is the circumference of its great circle. urface reas and Volumes of pheres 00 trand: Measurement opic: Measuring hsical ttributes center Geometr esson - ail otetaking Guide Vocabular and Ke oncepts. heorem -: reas and Volumes of imilar olids If the similarit ratio of two similar solids is a : b, then () the ratio of their corresponding areas is a : b, and () the ratio of their volumes is a : b. d circumference Hemispheres r sphere imilar solids have the same shape and all of their corresponding parts are proportional. he similarit ratio dimensions. r r diameter Great circle ame lass ate esson - esson bjective Find relationships between the ratios of the areas and volumes of similar solids of two similar objects is the ratio of their corresponding linear amples. Identifing imilar olids re the two solids similar? If so, give the similarit ratio. in. in. in. in. 00 trand: Measurement opic: stems of Measurement oth figures have the same shape. heck that the ratios of the corresponding dimensions are equal. reas and Volumes of imilar olids amples. Finding urface rea he circumference of a rubber ball is cm. alculate its surface area to the nearest whole number. tep First, find the radius. pr Use the formula for circumference. pr ubstitute for. r olve for r. p tep Use the radius to find the surface area... pr Use the formula for surface area of a sphere. p? ( ) ubstitute for r. p p implif. Use a calculator. o the nearest whole number, the surface area of the rubber ball is cm. Finding Volume Find the volume of the sphere. eave our answer in terms of π. V p p Use the formula for volume of a sphere.? ubstitute r 0. p implif. he volume of the sphere is 00p cm. Quick heck. 00 < r p.. Find the surface area of a sphere with d in. Give our answer in two was, in terms of π and rounded to the nearest square inch. p in., in.. Find the volume to the nearest cubic inch of a sphere with diameter 0 in.,0 in. ail otetaking Guide Geometr esson - ame lass ate Finding the imilarit atio Find the similarit ratio of two similar clinders with surface areas of π ft and π ft. Use the ratio of the surface areas to find the similarit ratios. a p he ratio of the surface areas is :. b a b p a b a b implif. ake the he similarit ratio is :. Quick heck. of each side.. re the two clinders similar? If so, give the similarit ratio. es; :. Find the similarit ratio of two similar prisms with surface areas m and m. : square root 0 cm 0 he ratio of the radii is, and the ratio of the heights is. he cones are not similar because. Geometr esson - ail otetaking Guide ail otetaking Guide Geometr esson - ll-in-ne nswers Version Geometr