BIG IDEAS MATH. A Bridge to Success. Ron Larson Laurie Boswell. Erie, Pennsylvania BigIdeasLearning.com

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1 IG I MH ridge to uccess on arson aurie oswell rie, ennsylvania igideasearning.com

2 5 ongruent riangles 5.1 ngles of riangles 5. ongruent olygons 5.3 roving riangle ongruence by 5.4 quilateral and Isosceles riangles 5.5 roving riangle ongruence by 5.6 roving riangle ongruence by and 5.7 Using ongruent riangles 5.8 oordinate roofs the ig Idea Hang Glider (p. 78) ifeguard ower (p. 55) arn (p. 48) Home ecor (p. 41) ainting (p. 35)

3 Maintaining Mathematical roficiency Using the Midpoint and istance ormulas xample 1 he endpoints of are (, 3) and (4, 7). ind the coordinates of the midpoint M. Use the Midpoint ormula. M ( , ) = M (, 10 ) = M(1, 5) he coordinates of the midpoint M are (1, 5). xample ind the distance between (0, 5) and (3, ). = (x x 1 ) + (y y 1 ) istance ormula = (3 0) + [ (5)] ubstitute. = ubtract. = valuate powers. = 58 dd. 7.6 Use a calculator. he distance between (0, 5) and (3, ) is about 7.6. ind the coordinates of the midpoint M of the segment with the given endpoints. hen find the distance between the two points. 1. ( 4, 1) and (0, 7). G(3, 6) and H(9, ) 3. U( 1, ) and V(8, 0) olving quations with Variables on oth ides xample 3 olve 5x = 3x. 5x = 3x Write the equation. +5x +5x dd 5x to each side. = x implify. = x olve the equation. 1 = x implify. he solution is x = 1. ivide each side by. 4. 7x + 1 = 3x t = t 6. 5p + 10 = 8p w + 13 = 11w x + 1 = 3 x 9. z = 4 + 9z 10. ONING Is it possible to find the length of a segment in a coordinate plane without using the istance ormula? xplain your reasoning. ynamic olutions available at igideasmath.com 9

4 Mathematical ractices efinitions, ostulates, and heorems ore oncept efinitions and iconditional tatements Mathematically profi cient students understand and use given defi nitions. definition is always an if and only if statement. Here is an example. efinition: wo geometric figures are congruent figures if and only if there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other. ecause this is a definition, it is a biconditional statement. It implies the following two conditional statements. 1. If two geometric figures are congruent figures, then there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other.. If there is a rigid motion or a composition of rigid motions that maps one geometric figure onto another, then the two geometric figures are congruent figures. efinitions, postulates, and theorems are the building blocks of geometry. In two-column proofs, the statements in the reason column are almost always definitions, postulates, or theorems. Monitoring rogress Identifying efinitions, ostulates, and heorems lassify each statement as a definition, a postulate, or a theorem. a. If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. b. If two coplanar lines have no point of intersection, then the lines are parallel. c. If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. OUION a. his is a theorem. It is the lternate Interior ngles onverse heorem (heorem 3.6) studied in ection 3.3. b. his is the definition of parallel lines. c. his is a postulate. It is the arallel ostulate (ostulate 3.1) studied in ection 3.1. In uclidean geometry, it is assumed, not proved, to be true. lassify each statement as a definition, a postulate, or a theorem. xplain your reasoning. 1. In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is 1.. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. 3. If two lines intersect to form a right angle, then the lines are perpendicular. 4. hrough any two points, there exists exactly one line. 30 hapter 5 ongruent riangles

5 5.1 ngles of riangles ssential uestion How are the angle measures of a triangle related? Writing a onjecture ONUING VI GUMN o be proficient in math, you need to reason inductively about data and write conjectures. Work with a partner. a. Use dynamic geometry software to draw any triangle and label it. b. ind the measures of the interior angles of the triangle. c. ind the sum of the interior angle measures. d. epeat parts (a) (c) with several other triangles. hen write a conjecture about the sum of the measures of the interior angles of a triangle. ample ngles m = m = m = Writing a onjecture Work with a partner. a. Use dynamic geometry software to draw any triangle and label it. b. raw an exterior angle at any vertex and find its measure. c. ind the measures of the two nonadjacent interior angles of the triangle. d. ind the sum of the measures of the two nonadjacent interior angles. ompare this sum to the measure of the exterior angle. e. epeat parts (a) (d) with several other triangles. hen write a conjecture that compares the measure of an exterior angle with the sum of the measures of the two nonadjacent interior angles. ample ngles m = m = m = ommunicate Your nswer 3. How are the angle measures of a triangle related? 4. n exterior angle of a triangle measures 3. What do you know about the measures of the interior angles? xplain your reasoning. ection 5.1 ngles of riangles 31

6 5.1 esson What You Will earn ore Vocabulary interior angles, p. 33 exterior angles, p. 33 corollary to a theorem, p. 35 revious triangle lassify triangles by sides and angles. ind interior and exterior angle measures of triangles. lassifying riangles by ides and by ngles ecall that a triangle is a polygon with three sides. You can classify triangles by sides and by angles, as shown below. ore oncept lassifying riangles by ides calene riangle Isosceles riangle quilateral riangle ING Notice that an equilateral triangle is also isosceles. n equiangular triangle is also acute. no congruent sides at least congruent sides 3 congruent sides lassifying riangles by ngles cute ight Obtuse quiangular riangle riangle riangle riangle 3 acute angles 1 right angle 1 obtuse angle 3 congruent angles lassify the triangular shape of the support beams in the diagram by its sides and by measuring its angles. lassifying riangles by ides and by ngles OUION he triangle has a pair of congruent sides, so it is isosceles. y measuring, the angles are 55, 55, and 70. o, it is an acute isosceles triangle. Monitoring rogress Help in nglish and panish at igideasmath.com 1. raw an obtuse isosceles triangle and an acute scalene triangle. 3 hapter 5 ongruent riangles

7 lassifying a riangle in the oordinate lane lassify O by its sides. hen determine whether it is a right triangle. ( 1, ) 4 y (6, 3) O(0, 0) x OUION tep 1 Use the istance ormula to find the side lengths. O = (x x 1 ) + (y y 1 ) = ( 1 0) + ( 0) = 5. O = (x x 1 ) + (y y 1 ) = (6 0) + (3 0) = = (x x 1 ) + (y y 1 ) = [6 ( 1)] + (3 ) = ecause no sides are congruent, O is a scalene triangle. tep heck for right angles. he slope of O is is 6 0 = 1 (. he product of the slopes is 1 ) O is a right angle. o, O is a right scalene triangle. Monitoring rogress =. he slope of O = 1. o, O O and Help in nglish and panish at igideasmath.com. has vertices (0, 0), (3, 3), and ( 3, 3). lassify the triangle by its sides. hen determine whether it is a right triangle. inding ngle Measures of riangles When the sides of a polygon are extended, other angles are formed. he original angles are the interior angles. he angles that form linear pairs with the interior angles are the exterior angles. heorem interior angles heorem 5.1 riangle um heorem he sum of the measures of the interior angles of a triangle is 180. exterior angles roof p. 34; x. 53, p. 38 m + m + m = 180 ection 5.1 ngles of riangles 33

8 o prove certain theorems, you may need to add a line, a segment, or a ray to a given diagram. n auxiliary line is used in the proof of the riangle um heorem. riangle um heorem Given rove m 1 + m + m 3 = lan for roof a. raw an auxiliary line through that is parallel to 1 3. b. how that m 4 + m + m 5 = 180, 1 4, and 3 5. c. y substitution, m 1 + m + m 3 = 180. lan in ction MN ON a. 1. raw parallel to. 1. arallel ostulate (ost. 3.1) b.. m 4 + m + m 5 = 180. ngle ddition ostulate (ost. 1.4) and definition of straight angle , lternate Interior ngles heorem (hm. 3.) 4. m l = m 4, m 3 = m 5 4. efinition of congruent angles c. 5. m l + m + m 3 = ubstitution roperty of quality heorem heorem 5. xterior ngle heorem he measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. roof x. 4, p m 1 = m + m ind m JM. OUION inding an ngle Measure tep 1 Write and solve an equation 70 (x 5) to find the value of x. (x 5) = 70 + x pply the xterior ngle heorem. M x = 75 olve for x. tep ubstitute 75 for x in x 5 to find m JM. x 5 = 75 5 = 145 o, the measure of JM is 145. J x 34 hapter 5 ongruent riangles

9 corollary to a theorem is a statement that can be proved easily using the theorem. he corollary below follows from the riangle um heorem. orollary orollary 5.1 orollary to the riangle um heorem he acute angles of a right triangle are complementary. roof x. 41, p. 37 m + m = 90 x x Modeling with Mathematics In the painting, the red triangle is a right triangle. he measure of one acute angle in the triangle is twice the measure of the other. ind the measure of each acute angle. OUION 1. Understand the roblem You are given a right triangle and the relationship between the two acute angles in the triangle. You need to find the measure of each acute angle.. Make a lan irst, sketch a diagram of the situation. You can use the orollary to the riangle um heorem and the given relationship between the two acute angles to write and solve an equation to find the measure of each acute angle. 3. olve the roblem et the measure of the smaller acute angle be x. hen the measure of the larger acute angle is x. he orollary to the riangle um heorem states that the acute angles of a right triangle are complementary. Use the corollary to set up and solve an equation. x + x = 90 orollary to the riangle um heorem x = 30 olve for x. o, the measures of the acute angles are 30 and (30 ) = ook ack dd the two angles and check that their sum satisfies the orollary to the riangle um heorem = 90 Monitoring rogress Help in nglish and panish at igideasmath.com 3. ind the measure of ind the measure of each acute angle. 3x x 40 1 (5x 10) (x 6) ection 5.1 ngles of riangles 35

10 5.1 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. WIING an a right triangle also be obtuse? xplain your reasoning.. OM H NN he measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles. Monitoring rogress and Modeling with Mathematics In xercises 3 6, classify the triangle by its sides and by measuring its angles. (ee xample 1.) In xercises 15 18, find the measure of the exterior angle. (ee xample 3.) 3. X 4. M (x ) Y Z N x J H (3x + 6) 4 (x + 18) In xercises 7 10, classify by its sides. hen determine whether it is a right triangle. (ee xample.) (, 3), (6, 3), (, 7) 8. (3, 3), (6, 9), (6, 3) 9. (1, 9), (4, 8), (, 5) (x + 8) (7x 16) 4x 10. (, 3), (0, 3), (3, ) In xercises 11 14, find m 1. hen classify the triangle by its angles In xercises 19, find the measure of each acute angle. (ee xample 4.) x (6x + 7) (11x ) x 0.. (3x + ) (19x 1) x (13x 5) 36 hapter 5 ongruent riangles

11 In xercises 3 6, find the measure of each acute angle in the right triangle. (ee xample 4.) 3. he measure of one acute angle is 5 times the measure of the other acute angle. 4. he measure of one acute angle is 8 times the measure of the other acute angle. 5. he measure of one acute angle is 3 times the sum of the measure of the other acute angle and he measure of one acute angle is twice the difference of the measure of the other acute angle and 1. O NYI In xercises 7 and 8, describe and correct the error in finding m m 1 = m 1 = m 1 = UING OO hree people are standing on a stage. he distances between the three people are shown in the diagram. lassify the triangle by its sides and by measuring its angles. 8 ft 6.5 ft 5 ft 38. UING UU Which of the following sets of angle measures could form a triangle? elect all that apply. 100, 50, 40 96, 74, , 113, 8 101, 41, 38 90, 45, 45 84, 6, MOING WIH MHMI You are bending a strip of metal into an isosceles triangle for a sculpture. he strip of metal is 0 inches long. he first bend is made 6 inches from one end. escribe two ways you could complete the triangle m = 180 m = 180 m 1 = HOUGH OVOING ind and draw an object (or part of an object) that can be modeled by a triangle and an exterior angle. escribe the relationship between the interior angles of the triangle and the exterior angle in terms of the object. 41. OVING OOY rove the orollary to the riangle um heorem (orollary 5.1). In xercises 9 36, find the measure of the numbered angle. Given is a right triangle. rove and are complementary OVING HOM rove the xterior ngle heorem (heorem 5.). Given, exterior rove m + m = m ection 5.1 ngles of riangles 37

12 43. II HINING Is it possible to draw an obtuse isosceles triangle? obtuse equilateral triangle? If so, provide examples. If not, explain why it is not possible. 44. II HINING Is it possible to draw a right isosceles triangle? right equilateral triangle? If so, provide an example. If not, explain why it is not possible. 45. MHMI ONNION is isosceles, = x, and = x 4. a. ind two possible values for x when the perimeter of is 3. b. How many possible values are there for x when the perimeter of is 1? 46. HOW O YOU I? lassify the triangles, in as many ways as possible, without finding any measurements. a. b. c. d. 48. MING N GUMN Your friend claims the measure of an exterior angle will always be greater than the sum of the nonadjacent interior angle measures. Is your friend correct? xplain your reasoning. MHMI ONNION In xercises 49 5, find the values of x and y y y 75 x x x y OVING HOM Use the diagram to write a proof of the riangle um heorem (heorem 5.1). Your proof should be different from the proof of the riangle um heorem shown in this lesson. y x 47. NYZING IONHI Which of the following could represent the measures of an exterior angle and two interior angles of a triangle? elect all that apply. 100, 6, 38 81, 57, 4 119, 68, 49 95, 85, , 78, , 101, 48 Maintaining Mathematical roficiency Use the diagram to find the measure of the segment or angle. (ection 1. and ection 1.5) 54. m H 55. m 56. GH 57. 5y 8 (6x + ) 3z + 6 eviewing what you learned in previous grades and lessons G 3y H 8z 9 (5x 7) (3x + 1) 38 hapter 5 ongruent riangles

13 5. ongruent olygons ssential uestion Given two congruent triangles, how can you use rigid motions to map one triangle to the other triangle? OOING O UU o be proficient in math, you need to look closely to discern a pattern or structure. escribing igid Motions Work with a partner. Of the four transformations you studied in hapter 4, which are rigid motions? Under a rigid motion, why is the image of a triangle always congruent to the original triangle? xplain your reasoning. ranslation eflection otation ilation inding a omposition of igid Motions Work with a partner. escribe a composition of rigid motions that maps to. Use dynamic geometry software to verify your answer. a. b c. d ommunicate Your nswer 3. Given two congruent triangles, how can you use rigid motions to map one triangle to the other triangle? 4. he vertices of are (1, 1), (3, ), and (4, 4). he vertices of are (, 1), (0, 0), and ( 1, ). escribe a composition of rigid motions that maps to. ection 5. ongruent olygons 39

14 5. esson What You Will earn ore Vocabulary corresponding parts, p. 40 revious congruent figures UY I Notice that both of the following statements are true. 1. If two triangles are congruent, then all their corresponding parts are congruent.. If all the corresponding parts of two triangles are congruent, then the triangles are congruent. Identify and use corresponding parts. Use the hird ngles heorem. Identifying and Using orresponding arts ecall that two geometric figures are congruent if and only if a rigid motion or a composition of rigid motions maps one of the figures onto the other. rigid motion maps each part of a figure to a corresponding part of its image. ecause rigid motions preserve length and angle measure, corresponding parts of congruent figures are congruent. In congruent polygons, this means that the corresponding sides and the corresponding angles are congruent. When is the image of after a rigid motion or a composition of rigid motions, you can write congruence statements for the corresponding angles and corresponding sides. orresponding angles orresponding sides,,,, When you write a congruence statement for two polygons, always list the corresponding vertices in the same order. You can write congruence statements in more than one way. wo possible congruence statements for the triangles above are or. When all the corresponding parts of two triangles are congruent, you can show that the triangles are congruent. Using the triangles above, first translate so that point maps to point. his translation maps to. Next, rotate counterclockwise through so that the image of coincides with. ecause, the rotation maps point to point. o, this rotation maps to. VIU ONING o help you identify corresponding parts, rotate. Now, reflect in the line through points and. his reflection maps the sides and angles of to the corresponding sides and corresponding angles of, so. o, to show that two triangles are congruent, it is sufficient to show that their corresponding parts are congruent. In general, this is true for all polygons. J Identifying orresponding arts Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. OUION he diagram indicates that J. orresponding angles J,, orresponding sides J,, J J 40 hapter 5 ongruent riangles

15 Using roperties of ongruent igures In the diagram, G. a. ind the value of x. b. ind the value of y. OUION a. You know that G. 8 ft (x 4) ft 10 (6y + x) b. You know that. G = m = m 1 = x 4 68 = (6y + x) 16 = x 68 = 6y = x 10 = y G ft howing hat igures re ongruent You divide the wall into orange and blue sections along J. Will the sections of the wall be the same size and shape? xplain. J 1 OUION rom the diagram, and 3 4 because all right angles are congruent. lso, by the ines erpendicular to a ransversal heorem (hm. 3.1),. hen 1 4 and 3 by the lternate Interior ngles heorem (hm. 3.). o, all pairs of corresponding angles are congruent. he diagram shows J, J, and. y the eflexive roperty of ongruence (hm..1), J J. o, all pairs of corresponding sides are congruent. ecause all corresponding parts are congruent, J J. Yes, the two sections will be the same size and shape. Monitoring rogress In the diagram, GH. 1. Identify all pairs of congruent corresponding parts.. ind the value of x. 3. In the diagram at the left, show that. Help in nglish and panish at igideasmath.com (4x + 5) H G UY I he properties of congruence that are true for segments and angles are also true for triangles. heorem heorem 5.3 roperties of riangle ongruence riangle congruence is reflexive, symmetric, and transitive. eflexive or any triangle,. ymmetric If, then. ransitive If and J, then J. roof igideasmath.com ection 5. ongruent olygons 41

16 Using the hird ngles heorem heorem heorem 5.4 hird ngles heorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. roof x. 19, p. 44 If and, then. Using the hird ngles heorem ind m. 45 N 30 OUION and, so by the hird ngles heorem,. y the riangle um heorem (heorem 5.1), m = = 105. o, m = m = 105 by the definition of congruent angles. roving hat riangles re ongruent Use the information in the figure to prove that. OUION Given,,, rove lan for roof a. Use the eflexive roperty of ongruence (hm..1) to show that. b. Use the hird ngles heorem to show that. lan in ction MN 1., a.. 3., ON 1. Given. eflexive roperty of ongruence (heorem.1) 3. Given b hird ngles heorem ll corresponding parts are congruent. Monitoring rogress Help in nglish and panish at igideasmath.com N Use the diagram. 4. ind m N. 5. What additional information is needed to conclude that N N? 4 hapter 5 ongruent riangles

17 5. xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. WIING ased on this lesson, what information do you need to prove that two triangles are congruent? xplain your reasoning.. IN WO, M UION Which is different? ind both answers. Is J? Is J? Is J? Is J? J Monitoring rogress and Modeling with Mathematics In xercises 3 and 4, identify all pairs of congruent corresponding parts. hen write another congruence statement for the polygons. (ee xample 1.) GHJ H G In xercises 5 8, XYZ MN. opy and complete the statement. J 10. MN U N 14 M 4 (x y) m 13 m (x 50) In xercises 11 and 1, show that the polygons are congruent. xplain your reasoning. (ee xample 3.) 11. W V Z X Y J N 1. X Y M U 5. m Y = X 6. m M = 7. m Z = 1 N Y Z M W In xercises 13 and 14, find m 1. (ee xample 4.) Z 8. XY = In xercises 9 and 10, find the values of x and y. (ee xample.) 9. GH 13. M Y 70 1 N X Z 14. (4y 4) 135 (10x + 65) H G 8 ection 5. ongruent olygons 43

18 15. OO riangular postage stamps, like the ones shown, are highly valued by stamp collectors. rove that. (ee xample 5.) 19. OVING HOM rove the hird ngles heorem (heorem 5.4) by using the riangle um heorem (heorem 5.1). 0. HOUGH OVOING raw a triangle. opy the triangle multiple times to create a rug design made of congruent triangles. Which property guarantees that all the triangles are congruent? Given,, is the midpoint of and. rove 16. OO Use the information in the figure to prove that G. O NYI In xercises 17 and 18, describe and correct the error Given XZY Y G X Z M 4 N Z m = m Z m = 4 MN because the corresponding angles are congruent. 1. ONING J is congruent to XYZ. Identify all pairs of congruent corresponding parts.. HOW O YOU I? In the diagram,. a. xplain how you know that and. b. xplain how you know that G G. c. xplain how you know that G G. d. o you have enough information to prove that G G? xplain. MHMI ONNION In xercises 3 and 4, use the given information to write and solve a system of linear equations to find the values of x and y. 3. MN, m = 40, m M = 90, m = (17x y), m = (x + 4y) 4. U XYZ, m = 8, m U = (4x + y), m X = 130, m Y = (8x 6y) 5. OO rove that the criteria for congruent triangles in this lesson is equivalent to the definition of congruence in terms of rigid motions. G Maintaining Mathematical roficiency What can you conclude from the diagram? (ection 1.6) 6. X V 7. Y Z U W N 8. eviewing what you learned in previous grades and lessons 9. I G J M H 44 hapter 5 ongruent riangles

19 5.3 roving riangle ongruence by ssential uestion What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent? rawing riangles Work with a partner. Use dynamic geometry software. UING OO GIY o be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data. a. onstruct circles with radii of units and 3 units centered at the origin. onstruct a 40 angle with its vertex at the origin. abel the vertex. b. ocate the point where one ray of the angle intersects the smaller circle and label this point. ocate the point where the other ray of the angle intersects the larger circle and label this point. hen draw. c. ind, m, and m. d. epeat parts (a) (c) several times, redrawing the angle in different positions. eep track of your results by copying and completing the table below. What can you conclude? m m m 1. (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) 3 40 ommunicate Your nswer. What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent? 3. How would you prove your conclusion in xploration 1(d)? ection 5.3 roving riangle ongruence by 45

20 5.3 esson ore Vocabulary revious congruent figures rigid motion UY I he included angle of two sides of a triangle is the angle formed by the two sides. What You Will earn Use the ide-ngle-ide () ongruence heorem. olve real-life problems. Using the ide-ngle-ide ongruence heorem heorem heorem 5.5 ide-ngle-ide () ongruence heorem If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If,, and, then. roof p. 46 ide-ngle-ide () ongruence heorem Given,, rove irst, translate so that point maps to point, as shown below. 46 hapter 5 ongruent riangles his translation maps to. Next, rotate counterclockwise through so that the image of coincides with, as shown below. ecause, the rotation maps point to point. o, this rotation maps to. Now, reflect in the line through points and, as shown below. ecause points and lie on, this reflection maps them onto themselves. ecause a reflection preserves angle measure and, the reflection maps to. ecause, the reflection maps point to point. o, this reflection maps to. ecause you can map to using a composition of rigid motions,.

21 Using the ongruence heorem UY I Make your proof easier to read by identifying the steps where you show congruent sides () and angles (). Write a proof. Given, rove OUION MN 1.. ON 1. Given. Given lternate Interior ngles heorem (hm. 3.) eflexive roperty of ongruence (hm..1) ongruence heorem Using and roperties of hapes In the diagram, and pass through the center M of the circle. What can you conclude about M and M? M OUION ecause they are vertical angles, M M. ll points on a circle are the same distance from the center, so M, M, M, and M are all congruent. o, M and M are congruent by the ongruence heorem. Monitoring rogress Help in nglish and panish at igideasmath.com In the diagram, is a square with four congruent sides and four right angles.,,, and U are the midpoints of the sides of. lso, U and V VU. V U 1. rove that V UV.. rove that U. ection 5.3 roving riangle ongruence by 47

22 opying a riangle Using onstruct a triangle that is congruent to using the ongruence heorem. Use a compass and straightedge. OUION tep 1 tep tep 3 tep 4 onstruct a side onstruct so that it is congruent to. onstruct an angle onstruct with vertex and side so that it is congruent to. onstruct a side onstruct so that it is congruent to. raw a triangle raw. y the ongruence heorem,. olving eal-ife roblems olving a eal-ife roblem You are making a canvas sign to hang on the triangular portion of the barn wall shown in the picture. You think you can use two identical triangular sheets of canvas. You know that and. Use the ongruence heorem to show that. OUION You are given that. y the eflexive roperty of ongruence (heorem.1),. y the definition of perpendicular lines, both and are right angles, so they are congruent. o, two pairs of sides and their included angles are congruent. and are congruent by the ongruence heorem. Monitoring rogress Help in nglish and panish at igideasmath.com 3. You are designing the window shown in the photo. You want to make congruent to G. You design the window so that G and G. Use the ongruence heorem to prove G. G 48 hapter 5 ongruent riangles

23 5.3 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. WIING What is an included angle?. OM H NN If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then. Monitoring rogress and Modeling with Mathematics In xercises 3 8, name the included angle between the pair of sides given. J In xercises 15 18, write a proof. (ee xample 1.) 15. Given bisects, rove 3. J and 4. and 5. and 6. J and J 7. and J 8. and 16. Given, rove In xercises 9 14, decide whether enough information is given to prove that the triangles are congruent using the ongruence heorem (heorem 5.5). xplain. 1 9., 10. MN, N M N 11. YXZ, WXZ 1. V, U 17. Given is the midpoint of and. rove Z W X Y V U 13. H, GH 14. M, MN 18. Given, rove G H N M ection 5.3 roving riangle ongruence by 49

24 In xercises 19, use the given information to name two triangles that are congruent. xplain your reasoning. (ee xample.) 19. U, and 0. is a square with is the center of four congruent sides and the circle. four congruent angles. U 1. UV is a regular. M MN, N, pentagon. and M and are centers of circles. V U 10 m M N 10 m ONUION In xercises 3 and 4, construct a triangle that is congruent to using the ongruence heorem (heorem 5.5) O NYI escribe and correct the error in finding the value of x. 4x + 6 X Y 5x 5 Z 5x 1 3x + 9 W 4x + 6 = 3x + 9 x + 6 = 9 x = 3 6. HOW O YOU I? What additional information do you need to prove that? 7. OO he Navajo rug is made of isosceles triangles. You know. Use the ongruence heorem (heorem 5.5) to show that. (ee xample 3.) 8. HOUGH OVOING here are six possible subsets of three sides or angles of a triangle:,,,,, and. Which of these correspond to congruence theorems? or those that do not, give a counterexample. 9. MHMI ONNION rove that. hen find the values 4y 6 of x and y. x + 6 3y MING N GUMN Your friend claims it is possible to construct a triangle congruent to by first constructing and, and then copying. Is your friend correct? xplain your reasoning. 31. OVING HOM rove the eflections in Intersecting ines heorem (heorem 4.3). 4x Maintaining Mathematical roficiency lassify the triangle by its sides and by measuring its angles. (ection 5.1) eviewing what you learned in previous grades and lessons hapter 5 ongruent riangles

25 5.4 quilateral and Isosceles riangles ssential uestion What conjectures can you make about the side lengths and angle measures of an isosceles triangle? Writing a onjecture about Isosceles riangles ONUING VI GUMN o be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures. Work with a partner. Use dynamic geometry software. a. onstruct a circle with a radius of 3 units centered at the origin. b. onstruct so that and are on the circle and is at the origin ample oints (0, 0) (.64, 1.4) ( 1.4,.64) egments = 3 = 3 = 4.4 ngles m = 90 m = 45 m = 45 c. ecall that a triangle is isosceles if it has at least two congruent sides. xplain why is an isosceles triangle. d. What do you observe about the angles of? e. epeat parts (a) (d) with several other isosceles triangles using circles of different radii. eep track of your observations by copying and completing the table below. hen write a conjecture about the angle measures of an isosceles triangle. m m m ample 1. (0, 0) (.64, 1.4) ( 1.4,.64) (0, 0) 3. (0, 0) 4. (0, 0) 5. (0, 0) f. Write the converse of the conjecture you wrote in part (e). Is the converse true? ommunicate Your nswer. What conjectures can you make about the side lengths and angle measures of an isosceles triangle? 3. How would you prove your conclusion in xploration 1(e)? in xploration 1(f)? ection 5.4 quilateral and Isosceles riangles 51

26 5.4 esson ore Vocabulary legs, p. 5 vertex angle, p. 5 base, p. 5 base angles, p. 5 What You Will earn Use the ase ngles heorem. Use isosceles and equilateral triangles. Using the ase ngles heorem triangle is isosceles when it has at least two congruent sides. When an isosceles triangle has exactly two congruent sides, these two sides are the legs. he angle formed by the legs is the vertex angle. he third side is the base of the isosceles triangle. he two angles adjacent to the base are called base angles. heorems heorem 5.6 ase ngles heorem If two sides of a triangle are congruent, then the angles opposite them are congruent. If, then. roof p. 5 vertex angle leg base angles base leg heorem 5.7 onverse of the ase ngles heorem If two angles of a triangle are congruent, then the sides opposite them are congruent. If, then. roof x. 7, p. 75 Given rove lan for roof ase ngles heorem a. raw so that it bisects. b. Use the ongruence heorem to show that. c. Use properties of congruent triangles to show that. lan in ction MN a. 1. raw, the angle bisector of. ON 1. onstruction of angle bisector.. efinition of angle bisector Given eflexive roperty of ongruence (hm..1) b ongruence heorem (hm. 5.5) c orresponding parts of congruent triangles are congruent. 5 hapter 5 ongruent riangles

27 Using the ase ngles heorem In,. Name two congruent angles. OUION, so by the ase ngles heorem,. Monitoring rogress opy and complete the statement. 1. If HG H, then.. If HJ JH, then. Help in nglish and panish at igideasmath.com G H J ecall that an equilateral triangle has three congruent sides. orollaries ING he corollaries state that a triangle is equilateral if and only if it is equiangular. orollary 5. orollary to the ase ngles heorem If a triangle is equilateral, then it is equiangular. roof x. 37, p. 58; x. 10, p. 353 orollary 5.3 orollary to the onverse of the ase ngles heorem If a triangle is equiangular, then it is equilateral. roof x. 39, p. 58 inding Measures in a riangle ind the measures of,, and. OUION he diagram shows that is equilateral. o, by the orollary to the ase ngles heorem, is equiangular. o, m = m = m. 3(m ) = 180 riangle um heorem (heorem 5.1) m = 60 ivide each side by 3. U 5 he measures of,, and are all 60. Monitoring rogress 3. ind the length of for the triangle at the left. Help in nglish and panish at igideasmath.com ection 5.4 quilateral and Isosceles riangles 53

28 Using Isosceles and quilateral riangles onstructing an quilateral riangle onstruct an equilateral triangle that has side lengths congruent to. Use a compass and straightedge. OUION tep 1 tep tep 3 tep 4 opy a segment opy. raw an arc raw an arc with center and radius. raw an arc raw an arc with center and radius. abel the intersection of the arcs from teps and 3 as. raw a triangle raw. ecause and are radii of the same circle,. ecause and are radii of the same circle,. y the ransitive roperty of ongruence (heorem.1),. o, is equilateral. ind the values of x and y in the diagram. Using Isosceles and quilateral riangles y 4 x + 1 OMMON O You cannot use N to refer to NM because three angles have N as their vertex. N M OUION tep 1 ind the value of y. ecause N is equiangular, it is also equilateral and N. o, y = 4. tep ind the value of x. ecause NM MN, N M, and MN is isosceles. You also know that N = 4 because N is equilateral. N = M efinition of congruent segments 4 = x + 1 ubstitute 4 for N and x + 1 for M. 3 = x ubtract 1 from each side. 54 hapter 5 ongruent riangles

29 olving a Multi-tep roblem In the lifeguard tower, and a. xplain how to prove that. b. xplain why is isosceles. OMMON O When you redraw the triangles so that they do not overlap, be careful to copy all given information and labels correctly. OUION a. raw and label and so that they do not overlap. You can see that,, and. o, by the ongruence heorem (heorem 5.5), b. rom part (a), you know that 1 because corresponding parts of congruent triangles are congruent. y the onverse of the ase ngles heorem,, and is isosceles. Monitoring rogress 4. ind the values of x and y in the diagram. Help in nglish and panish at igideasmath.com y x 5. In xample 4, show that. ection 5.4 quilateral and Isosceles riangles 55

30 5.4 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. VOUY escribe how to identify the vertex angle of an isosceles triangle.. WIING What is the relationship between the base angles of an isosceles triangle? xplain. Monitoring rogress and Modeling with Mathematics In xercises 3 6, copy and complete the statement. tate which theorem you used. (ee xample 1.) 3. If, then. 4. If, then. 1. MOING WIH MHMI logo in an advertisement is an equilateral triangle with a side length of 7 centimeters. ketch the logo and give the measure of each side. In xercises 13 16, find the values of x and y. (ee xample 3.) 13. y x 14. y 40 x 5. If, then. 6. If, then. In xercises 7 10, find the value of x. (ee xample.) 15. 8y x x 1 x x M x 11. MOING WIH MHMI he dimensions of a sports pennant are given in the diagram. ind the values of x and y. 79 x W 5 y N 16. 3x 5 y + 1 5y 4 ONUION In xercises 17 and 18, construct an equilateral triangle whose sides are the given length inches inches 19. O NYI escribe and correct the error in finding the length of. 5 6 ecause,. o, = hapter 5 ongruent riangles

31 0. OM OVING he diagram represents part of the exterior of the ow ower in algary, lberta, anada. In the diagram, and are congruent equilateral triangles. (ee xample 4.) a. xplain why is isosceles. b. xplain why. c. how that and are congruent. d. ind the measure of. 1. INING N In the pattern shown, each small triangle is an equilateral triangle with an area of 1 square unit. a. xplain how you know that any triangle made out of equilateral triangles is equilateral. b. ind the areas of the first four triangles in the pattern. c. escribe any patterns in the areas. redict the area of the seventh triangle in the pattern. xplain your reasoning.. ONING he base of isosceles XYZ is YZ. What can you prove? elect all that apply. XY XZ X Y Y Z YZ ZX In xercises 3 and 4, find the perimeter of the triangle in. (4x + 1) in. (x + 4) in. riangle (1 x) in. rea 1 square unit (x 3) in. (x + 5) in. MOING WIH MHMI In xercises 5 8, use the diagram based on the color wheel. he 1 triangles in the diagram are isosceles triangles with congruent vertex angles. green blue bluepurple bluegreen yellowgreen redpurple yellow purple redorange orange yelloworange red 5. omplementary colors lie directly opposite each other on the color wheel. xplain how you know that the yellow triangle is congruent to the purple triangle. 6. he measure of the vertex angle of the yellow triangle is 30. ind the measures of the base angles. 7. race the color wheel. hen form a triangle whose vertices are the midpoints of the bases of the red, yellow, and blue triangles. (hese colors are the primary colors.) What type of triangle is this? 8. Other triangles can be formed on the color wheel that are congruent to the triangle in xercise 7. he colors on the vertices of these triangles are called triads. What are the possible triads? 9. II HINING re isosceles triangles always acute triangles? xplain your reasoning. 30. II HINING Is it possible for an equilateral triangle to have an angle measure other than 60? xplain your reasoning. 31. MHMI ONNION he lengths of the sides of a triangle are 3t, 5t 1, and t + 0. ind the values of t that make the triangle isosceles. xplain your reasoning. 3. MHMI ONNION he measure of an exterior angle of an isosceles triangle is x. Write expressions representing the possible angle measures of the triangle in terms of x. 33. WIING xplain why the measure of the vertex angle of an isosceles triangle must be an even number of degrees when the measures of all the angles of the triangle are whole numbers. ection 5.4 quilateral and Isosceles riangles 57

32 34. OM OVING he triangular faces of the peaks on a roof are congruent isosceles triangles with vertex angles U and V. 37. OVING OOY rove that the orollary to the ase ngles heorem (orollary 5.) follows from the ase ngles heorem (heorem 5.6). 6.5 m U V 38. HOW O YOU I? You are designing fabric purses to sell at the school fair. W 8 m X Y 100 a. Name two angles congruent to WUX. xplain your reasoning. b. ind the distance between points U and V. 35. OM OVING boat is traveling parallel to the shore along. When the boat is at point, the captain measures the angle to the lighthouse as 35. fter the boat has traveled.1 miles, the captain measures the angle to the lighthouse to be mi a. ind. xplain your reasoning. b. xplain how to find the distance between the boat and the shoreline. 36. HOUGH OVOING he postulates and theorems in this book represent uclidean geometry. In spherical geometry, all points are points on the surface of a sphere. line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, do all equiangular triangles have the same angle measures? Justify your answer. a. xplain why. b. Name the isosceles triangles in the purse. c. Name three angles that are congruent to. 39. OVING OOY rove that the orollary to the onverse of the ase ngles heorem (orollary 5.3) follows from the onverse of the ase ngles heorem (heorem 5.7). 40. MING N GUMN he coordinates of two points are (0, 6) and U(6, 0). Your friend claims that points, U, and V will always be the vertices of an isosceles triangle when V is any point on the line y = x. Is your friend correct? xplain your reasoning. 41. OO Use the diagram to prove that is equilateral. Given is equilateral. rove is equilateral. Maintaining Mathematical roficiency eviewing what you learned in previous grades and lessons Use the given property to complete the statement. (ection.5) 4. eflexive roperty of ongruence (heorem.1): 43. ymmetric roperty of ongruence (heorem.1): If, then J. 44. ransitive roperty of ongruence (heorem.1): If, and UV, then. 58 hapter 5 ongruent riangles

33 What id You earn? ore Vocabulary interior angles, p. 33 exterior angles, p. 33 corollary to a theorem, p. 35 corresponding parts, p. 40 legs (of an isosceles triangle), p. 5 vertex angle (of an isosceles triangle), p. 5 base (of an isosceles triangle), p. 5 base angles (of an isosceles triangle), p. 5 ore oncepts lassifying riangles by ides, p. 3 lassifying riangles by ngles, p. 3 heorem 5.1 riangle um heorem, p. 33 heorem 5. xterior ngle heorem, p. 34 orollary 5.1 orollary to the riangle um heorem, p. 35 Identifying and Using orresponding arts, p. 40 heorem 5.3 roperties of riangle ongruence, p. 41 heorem 5.4 hird ngles heorem, p. 4 heorem 5.5 ide-ngle-ide () ongruence heorem, p. 46 heorem 5.6 ase ngles heorem, p. 5 heorem 5.7 onverse of the ase ngles heorem, p. 5 orollary 5. orollary to the ase ngles heorem, p. 53 orollary 5.3 orollary to the onverse of the ase ngles heorem, p. 53 Mathematical ractices 1. In xercise 37 on page 37, what are you given? What relationships are present? What is your goal?. xplain the relationships present in xercise 3 on page escribe at least three different patterns created using triangles for the picture in xercise 0 on page 57. tudy kills Visual earners raw a picture of a word problem. raw a picture of a word problem before starting to solve the problem. You do not have to be an artist. When making a review card for a word problem, include a picture. his will help you recall the information while taking a test. Make sure your notes are visually neat for easy recall. 59

34 uiz ind the measure of the exterior angle. (ection 5.1) (5x + ) y x 6x (15x + 34) (1x + 6) Identify all pairs of congruent corresponding parts. hen write another congruence statement for the polygons. (ection 5.) WXYZ Y X W Z ecide whether enough information is given to prove that the triangles are congruent using the ongruence heorem (hm. 5.5). If so, write a proof. If not, explain why. (ection 5.3) 6., 7. GH, HJ 8. M, NM G M H J N opy and complete the statement. tate which theorem you used. (ection 5.4) 9. If VW WX, then. 10. If XZ XY, then. W X 11. If ZVX ZXV, then. 1. If XYZ ZXY, then. V ind the values of x and y. (ection 5. and ection 5.4) x 1 13 (5y 7) ft ft 6y Z Y (x + ) In a right triangle, the measure of one acute angle is 4 times the difference of the measure of the other acute angle and 5. ind the measure of each acute angle in the triangle. (ection 5.1) 16. he figure shows a stained glass window. (ection 5.1 and ection 5.3) a. lassify triangles 1 4 by their angles. b. lassify triangles 4 6 by their sides. c. Is there enough information given to prove that 7 8? If so, label the vertices and write a proof. If not, determine what additional information is needed. 60 hapter 5 ongruent riangles

35 5.5 roving riangle ongruence by ssential uestion What can you conclude about two triangles when you know the corresponding sides are congruent? rawing riangles Work with a partner. Use dynamic geometry software. a. onstruct circles with radii of units and 3 units centered at the origin. abel the origin. hen draw of length 4 units. 4 3 UING OO GIY o be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data. b. Move so that is on the smaller circle and is on the larger circle. hen draw. c. xplain why the side lengths of are, 3, and 4 units. d. ind m, m, and m. e. epeat parts (b) and (d) several times, moving to different locations. eep track of your results by copying and completing the table below. What can you conclude? m m m 1. (0, 0) 3 4. (0, 0) (0, 0) (0, 0) (0, 0) 3 4 ommunicate Your nswer. What can you conclude about two triangles when you know the corresponding sides are congruent? 3. How would you prove your conclusion in xploration 1(e)? ection 5.5 roving riangle ongruence by 61

36 5.5 esson ore Vocabulary legs, p. 64 hypotenuse, p. 64 revious congruent figures rigid motion What You Will earn Use the ide-ide-ide () ongruence heorem. Use the Hypotenuse-eg (H) ongruence heorem. Using the ide-ide-ide ongruence heorem heorem heorem 5.8 ide-ide-ide () ongruence heorem If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If,, and, then. ide-ide-ide () ongruence heorem Given,, rove irst, translate so that point maps to point, as shown below. his translation maps to. Next, rotate counterclockwise through so that the image of coincides with, as shown below ecause, the rotation maps point to point. o, this rotation maps to. raw an auxiliary line through points and. his line creates 1,, 3, and 4, as shown at the left. ecause, is an isosceles triangle. ecause, is an isosceles triangle. y the ase ngles heorem (hm. 5.6), 1 3 and 4. y the definition of congruence, m 1 = m 3 and m = m 4. y construction, m = m 1 + m and m = m 3 + m 4. You can now use the ubstitution roperty of quality to show m = m. m = m 1 + m ngle ddition ostulate (ostulate 1.4) = m 3 + m 4 ubstitute m 3 for m 1 and m 4 for m. = m ngle ddition ostulate (ostulate 1.4) y the definition of congruence,. o, two pairs of sides and their included angles are congruent. y the ongruence heorem (hm. 5.5),. o, a composition of rigid motions maps to. ecause a composition of rigid motions maps to and a composition of rigid motions maps to, a composition of rigid motions maps to. o,. 6 hapter 5 ongruent riangles

37 Using the ongruence heorem Write a proof. N, M NM Given N rove M NM M OUION MN ON 1. Given 1. N. M NM 3. M M. Given 3. eflexive roperty of ongruence (hm..1) 4. M NM 4. ongruence heorem Monitoring rogress Help in nglish and panish at igideasmath.com ecide whether the congruence statement is true. xplain your reasoning. 1. G HJ. J 3 G H olving a eal-ife roblem xplain why the bench with the diagonal support is stable, while the one without the support can collapse. OUION he bench with the diagonal support forms triangles with fixed side lengths. y the ongruence heorem, these triangles cannot change shape, so the bench is stable. he bench without the diagonal support is not stable because there are many possible quadrilaterals with the given side lengths. Monitoring rogress Help in nglish and panish at igideasmath.com etermine whether the figure is stable. xplain your reasoning ection 5.5 hs_geo_pe_0505.indd roving riangle ongruence by 63 1/19/15 10:30 M

38 opying a riangle Using onstruct a triangle that is congruent to using the ongruence heorem. Use a compass and straightedge. OUION tep 1 tep tep 3 tep 4 onstruct a side onstruct so that it is congruent to. raw an arc Open your compass to the length. Use this length to draw an arc with center. raw an arc raw an arc with radius and center that intersects the arc from tep. abel the intersection point. raw a triangle raw. y the ongruence heorem,. Using the Hypotenuse-eg ongruence heorem You know that and are valid methods for proving that triangles are congruent. What about? In general, is not a valid method for proving that triangles are congruent. In the triangles below, two pairs of sides and a pair of angles not included between them are congruent, but the triangles are not congruent. leg hypotenuse leg While is not valid in general, there is a special case for right triangles. In a right triangle, the sides adjacent to the right angle are called the legs. he side opposite the right angle is called the hypotenuse of the right triangle. heorem heorem 5.9 Hypotenuse-eg (H) ongruence heorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. If,, and m = m = 90, then. roof x. 38, p. 470; igideasmath.com 64 hapter 5 ongruent riangles

39 UY I If you have trouble matching vertices to letters when you separate the overlapping triangles, leave the triangles in their original orientations. W Z Y Z X Y Write a proof. Given WY XZ, WZ ZY, XY Z Y rove OUION WYZ XZY Using the Hypotenuse-eg ongruence heorem edraw the triangles so they are side by side with corresponding parts in the same position. Mark the given information in the diagram. MN H 1. WY XZ. WZ ZY, XY ZY ON 1. Given. Given 3. Z and Y are right angles. 3. efinition of lines 4. WYZ and XZY are right triangles. 4. efinition of a right triangle 5. ZY YZ 5. eflexive roperty of ongruence (hm..1) 6. WYZ XZY 6. H ongruence heorem W Z W Z Y X Y X Y Z Using the Hypotenuse-eg ongruence heorem he television antenna is perpendicular to the plane containing points,,, and. ach of the cables running from the top of the antenna to,, and has the same length. rove that,, and are congruent. Given,,, rove OUION You are given that and. o, and are right angles by the definition of perpendicular lines. y definition, and are right triangles. You are given that the hypotenuses of these two triangles, and, are congruent. lso, is a leg for both triangles, and by the eflexive roperty of ongruence (hm..1). o, by the Hypotenuse-eg ongruence heorem,. You can use similar reasoning to prove that. o, by the ransitive roperty of riangle ongruence (hm. 5.3),. Monitoring rogress Use the diagram. Help in nglish and panish at igideasmath.com 7. edraw and side by side with corresponding parts in the same position. 8. Use the information in the diagram to prove that. ection 5.5 roving riangle ongruence by 65

40 5.5 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. OM H NN he side opposite the right angle is called the of the right triangle.. WHIH ON ON ONG? Which triangle s legs do not belong with the other three? xplain your reasoning. Monitoring rogress and Modeling with Mathematics In xercises 3 and 4, decide whether enough information is given to prove that the triangles are congruent using the ongruence heorem (heorem 5.8). xplain. 3., 4., 9. G 10. J JM G J M In xercises 5 and 6, decide whether enough information is given to prove that the triangles are congruent using the H ongruence heorem (heorem 5.9). xplain. 5., 6., In xercises 7 10, decide whether the congruence statement is true. xplain your reasoning. (ee xample 1.) In xercises 11 and 1, determine whether the figure is stable. xplain your reasoning. (ee xample.) In xercises 13 and 14, redraw the triangles so they are side by side with corresponding parts in the same position. hen write a proof. (ee xample 3.) 13. Given,, rove 14. Given G is the midpoint of H, G GI, and H are right angles. rove G HIG G 66 hapter 5 ongruent riangles I H

41 In xercises 15 and 16, write a proof. 15. Given M J, MJ rove MJ J. MOING WIH MHMI he distances between consecutive bases on a softball field are the same. he distance from home plate to second base is the same as the distance from first base to third base. he angles created at each base are 90. rove H H. (ee xample 4.) J M second base () 16. Given WX VZ, WY VY, YZ YX rove VWX WVZ third base () first base () W X V Y Z home plate (H) ONUION In xercises 17 and 18, construct a triangle that is congruent to using the ongruence heorem (heorem 5.8). 3. ONING o support a tree, you attach wires from the trunk of the tree to stakes in the ground, as shown in the diagram O NYI escribe and correct the error in identifying congruent triangles. U X Z V UV XYZ by the ongruence heorem. 0. O NYI escribe and correct the error in determining the value of x that makes the triangles congruent. x + 1 J 6x M 4x + 4 3x 1 Y 6x = x + 1 4x = 1 x = MING N GUMN Your friend claims that in order to use the ongruence heorem (heorem 5.8) to prove that two triangles are congruent, both triangles must be equilateral triangles. Is your friend correct? xplain your reasoning. J a. What additional information do you need to use the H ongruence heorem (heorem 5.9) to prove that J M? b. uppose is the midpoint of JM. Name a theorem you could use to prove that J M. xplain your reasoning. 4. ONING Use the photo of the Navajo rug, where and. M a. What additional information do you need to use the ongruence heorem (heorem 5.8) to prove that? b. What additional information do you need to use the H ongruence heorem (heorem 5.9) to prove that? ection 5.5 roving riangle ongruence by 67

42 In xercises 5 8, use the given coordinates to determine whether. 5. (, ), (4, ), (4, 6), (5, 7), (5, 1), (13, 1) 6. (, 1), (3, 3), (7, 5), (3, 6), (8, ), (10, 11) 7. (0, 0), (6, 5), (9, 0), (0, 1), (6, 6), (9, 1) 8. ( 5, 7), ( 5, ), (0, ), (0, 6), (0, 1), (4, 1) 9. II HINING You notice two triangles in the tile floor of a hotel lobby. You want to determine whether the triangles are congruent, but you only have a piece of string. an you determine whether the triangles are congruent? xplain. 30. HOW O YOU I? here are several theorems you can use to show that the triangles in the square pattern are congruent. Name two of them. 3. HOUGH OVOING he postulates and theorems in this book represent uclidean geometry. In spherical geometry, all points are points on the surface of a sphere. line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, do you think that two triangles are congruent if their corresponding sides are congruent? Justify your answer. UING OO In xercises 33 and 34, use the given information to sketch MN and U. Mark the triangles with the given information. 33. M MN, U, M NM U 34. M MN, U, M, N U 35. II HINING he diagram shows the light created by two spotlights. oth spotlights are the same distance from the stage. G 31. MING N GUMN Your cousin says that J is congruent to MJ by the ongruence heorem (hm. 5.8). Your friend says that J is congruent to MJ by the H ongruence heorem (hm. 5.9). Who is correct? xplain your reasoning. J a. how that. tate which theorem or postulate you used and explain your reasoning. b. re all four right triangles shown in the diagram congruent? xplain your reasoning. 36. MHMI ONNION ind all values of x that make the triangles congruent. xplain. 5x 5x 4x + 3 3x + 10 M Maintaining Mathematical roficiency Use the congruent triangles. (ection 5.) 37. Name the segment in that is congruent to. 38. Name the segment in that is congruent to. 39. Name the angle in that is congruent to. 40. Name the angle in that is congruent to. eviewing what you learned in previous grades and lessons 68 hapter 5 ongruent riangles

43 5.6 roving riangle ongruence by and ssential uestion What information is sufficient to determine whether two triangles are congruent? etermining Whether Is ufficient Work with a partner. a. Use dynamic geometry software to construct. onstruct the triangle so that vertex is at the origin, has a length of 3 units, and has a length of units. b. onstruct a circle with a radius of units centered at the origin. ocate point where the circle intersects. raw ample oints (0, 3) (0, 0) (, 0) (0.77, 1.85) egments = 3 = 3.61 = = 1.38 ngle m = ONUING VI GUMN o be proficient in math, you need to recognize and use counterexamples. c. and have two congruent sides and a nonincluded congruent angle. Name them. d. Is? xplain your reasoning. e. Is sufficient to determine whether two triangles are congruent? xplain your reasoning. etermining Valid ongruence heorems Work with a partner. Use dynamic geometry software to determine which of the following are valid triangle congruence theorems. or those that are not valid, write a counterexample. xplain your reasoning. ossible ongruence heorem Valid or not valid? ommunicate Your nswer 3. What information is sufficient to determine whether two triangles are congruent? 4. Is it possible to show that two triangles are congruent using more than one congruence theorem? If so, give an example. ection 5.6 roving riangle ongruence by and 69

44 5.6 esson ore Vocabulary revious congruent figures rigid motion What You Will earn Use the and ongruence heorems. Using the and ongruence heorems heorem heorem 5.10 ngle-ide-ngle () ongruence heorem If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. If,, and, then. roof p. 70 ngle-ide-ngle () ongruence heorem Given,, rove irst, translate so that point maps to point, as shown below. 70 hapter 5 ongruent riangles his translation maps to. Next, rotate counterclockwise through so that the image of coincides with, as shown below. ecause, the rotation maps point to point. o, this rotation maps to. Now, reflect in the line through points and, as shown below. ecause points and lie on, this reflection maps them onto themselves. ecause a reflection preserves angle measure and, the reflection maps to. imilarly, because, the reflection maps to. he image of lies on and. ecause and only have point in common, the image of must be. o, this reflection maps to. ecause you can map to using a composition of rigid motions,.

45 heorem heorem 5.11 ngle-ngle-ide () ongruence heorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. If,, and, then. roof p. 71 ngle-ngle-ide () ongruence heorem Given,, rove You are given and. y the hird ngles heorem (heorem 5.4),. You are given. o, two pairs of angles and their included sides are congruent. y the ongruence heorem,. Identifying ongruent riangles an the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use. a. b. c. OMMON O You need at least one pair of congruent corresponding sides to prove two triangles are congruent. OUION a. he vertical angles are congruent, so two pairs of angles and a pair of non-included sides are congruent. he triangles are congruent by the ongruence heorem. b. here is not enough information to prove the triangles are congruent, because no sides are known to be congruent. c. wo pairs of angles and their included sides are congruent. he triangles are congruent by the ongruence heorem. Monitoring rogress 1. an the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use. Help in nglish and panish at igideasmath.com W X Z Y ection 5.6 roving riangle ongruence by and 71

46 opying a riangle Using onstruct a triangle that is congruent to using the ongruence heorem. Use a compass and straightedge. OUION tep 1 tep tep 3 tep 4 onstruct a side onstruct so that it is congruent to. onstruct an angle onstruct with vertex and side so that it is congruent to. onstruct an angle onstruct with vertex and side so that it is congruent to. abel a point abel the intersection of the sides of and that you constructed in teps and 3 as. y the ongruence heorem,. Using the ongruence heorem Write a proof. Given, rove OUION MN ON 1. Given. lternate Interior ngles heorem (hm. 3.) 3. Given 4. Vertical ngles ongruence heorem (hm.6) ongruence heorem Monitoring rogress Help in nglish and panish at igideasmath.com. In the diagram,,, and. rove. 7 hapter 5 ongruent riangles

47 Using the ongruence heorem Write a proof. Given H G, and are right angles. rove HG GH OUION H G MN 1. H G ON 1. Given. GH HG. lternate Interior ngles heorem (heorem 3.) 3. and are right angles. 3. Given HG GH 4. ight ngles ongruence heorem (heorem.3) 5. eflexive roperty of ongruence (heorem.1) 6. HG GH 6. ongruence heorem Monitoring rogress Help in nglish and panish at igideasmath.com 3. In the diagram, U and VU. rove VU. U V oncept ummary riangle ongruence heorems You have learned five methods for proving that triangles are congruent. H (right s only) wo sides and the included angle are congruent. ll three sides are congruent. he hypotenuse and one of the legs are congruent. wo angles and the included side are congruent. wo angles and a non-included side are congruent. In the xercises, you will prove three additional theorems about the congruence of right triangles: Hypotenuse-ngle, eg-eg, and ngle-eg. ection 5.6 roving riangle ongruence by and 73

48 5.6 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. WIING How are the ongruence heorem (heorem 5.11) and the ongruence heorem (heorem 5.10) similar? How are they different?. WIING You know that a pair of triangles has two pairs of congruent corresponding angles. What other information do you need to show that the triangles are congruent? Monitoring rogress and Modeling with Mathematics In xercises 3 6, decide whether enough information is given to prove that the triangles are congruent. If so, state the theorem you would use. (ee xample 1.) 3., 4., 5. XYZ, J 6. V, UV Y Z V In xercises 9 1, decide whether you can use the given information to prove that. xplain your reasoning. 9.,, 10.,, 11.,, 1.,, ONUION In xercises 13 and 14, construct a triangle that is congruent to the given triangle using the ongruence heorem (heorem 5.10). Use a compass and straightedge J X J U In xercises 7 and 8, state the third congruence statement that is needed to prove that GH MN using the given theorem. G M O NYI In xercises 15 and 16, describe and correct the error. 15. J G H J HG by the ongruence heorem. 7. Given GH MN, G M, H Use the ongruence heorem (hm. 5.11). 8. Given G M, G M, N 16. X W V VWX by the ongruence heorem. Use the ongruence heorem (hm. 5.10). 74 hapter 5 ongruent riangles

49 OO In xercises 17 and 18, prove that the triangles are congruent using the ongruence heorem (heorem 5.10). (ee xample.) 17. Given. M is the midpoint of N N, N M, M N and a leg of a right triangle are congruent to an angle and a leg of a second right triangle, then the triangles are congruent. 4. ONING What additional information do you need to prove J MN by the ongruence heorem (heorem 5.10)? NM M rove 3. ngle-eg () ongruence heorem If an angle M J H NH N M H M J N J NM M J 18. Given J, J J, rove 5. MHMI ONNION his toy J contains and. an you conclude that from the given angle measures? xplain. J OO In xercises 19 and 0, prove that the triangles are congruent using the ongruence heorem (heorem 5.11). (ee xample 3.) m = (8x 3) 19. Given VW UW, X Z m = (4y 4) rove m = (5x + 10) XWV ZWU Z V m = (3y + ) X Y m = (x 8) U m = (y 6) W 6. ONING Which of the following congruence 0. Given rove statements are true? elect all that apply. NM M, N U UV NM M V XVW N W X V VWU M OO In xercises 1 3, write a paragraph proof for the theorem about right triangles. 1. Hypotenuse-ngle (H) ongruence heorem If an angle and the hypotenuse of a right triangle are congruent to an angle and the hypotenuse of a second right triangle, then the triangles are congruent.. eg-eg () ongruence heorem If the legs of a right triangle are congruent to the legs of a second right triangle, then the triangles are congruent. ection 5.6 hs_geo_pe_0506.indd 75 V VUW U V 7. OVING HOM rove the onverse of the ase ngles heorem (heorem 5.7). (Hint: raw an auxiliary line inside the triangle.) 8. MING N GUMN Your friend claims to be able to rewrite any proof that uses the ongruence heorem (hm. 5.11) as a proof that uses the ongruence heorem (hm. 5.10). Is this possible? xplain your reasoning. roving riangle ongruence by and 75 1//15 8:06 M

50 9. MOING WIH MHMI When a light ray from an object meets a mirror, it is reflected back to your eye. or example, in the diagram, a light ray from point is reflected at point and travels back to point. he law of refl ection states that the angle of incidence,, is congruent to the angle of reflection,. a. rove that is congruent to. Given, rove b. Verify that is isosceles. c. oes moving away from the mirror have any effect on the amount of his or her reflection a person sees? xplain. 30. HOW O YOU I? Name as many pairs of congruent triangles as you can from the diagram. xplain how you know that each pair of triangles is congruent. 31. ONUION onstruct a triangle. how that there is no congruence rule by constructing a second triangle that has the same angle measures but is not congruent. 3. HOUGH OVOING Graph theory is a branch of mathematics that studies vertices and the way they are connected. In graph theory, two polygons are isomorphic if there is a one-to-one mapping from one polygon s vertices to the other polygon s vertices that preserves adjacent vertices. In graph theory, are any two triangles isomorphic? xplain your reasoning. 33. MHMI ONNION ix statements are given about UV and XYZ. U XY UV YZ V XZ X U Y V Z U V Z X a. ist all combinations of three given statements that would provide enough information to prove that UV is congruent to XYZ. b. You choose three statements at random. What is the probability that the statements you choose provide enough information to prove that the triangles are congruent? Y Maintaining Mathematical roficiency ind the coordinates of the midpoint of the line segment with the given endpoints. (ection 1.3) 34. (1, 0) and (5, 4) 35. J(, 3) and (4, 1) 36. ( 5, 7) and (, 4) opy the angle using a compass and straightedge. (ection 1.5) eviewing what you learned in previous grades and lessons hapter 5 ongruent riangles

51 5.7 Using ongruent riangles ssential uestion How can you use congruent triangles to make an indirect measurement? Measuring the Width of a iver IIUING H ONING O OH o be proficient in math, you need to listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Work with a partner. he figure shows how a surveyor can measure the width of a river by making measurements on only one side of the river. a. tudy the figure. hen explain how the surveyor can find the width of the river. b. Write a proof to verify that the method you described in part (a) is valid. Given is a right angle, is a right angle, c. xchange proofs with your partner and discuss the reasoning used. Measuring the Width of a iver Work with a partner. It was reported that one of Napoleon s officers estimated the width of a river as follows. he officer stood on the bank of the river and lowered the visor on his cap until the farthest thing visible was the edge of the bank on the other side. He then turned and noted the point on his side that was in line with the tip of his visor and his eye. he officer then paced the distance to this point and concluded that distance was the width of the river. a. tudy the figure. hen explain how the officer concluded that the width of the river is G. b. Write a proof to verify that the conclusion the officer made is correct. Given G is a right angle, is a right angle, G c. xchange proofs with your partner and discuss the reasoning used. G ommunicate Your nswer 3. How can you use congruent triangles to make an indirect measurement? 4. Why do you think the types of measurements described in xplorations 1 and are called indirect measurements? ection 5.7 Using ongruent riangles 77

52 5.7 esson ore Vocabulary revious congruent figures corresponding parts construction What You Will earn Use congruent triangles. rove constructions. Using ongruent riangles ongruent triangles have congruent corresponding parts. o, if you can prove that two triangles are congruent, then you know that their corresponding parts must be congruent as well. Using ongruent riangles xplain how you can use the given information to prove that the hang glider parts are congruent. 1 Given 1, rove OUION If you can show that, then you will know that. irst, copy the diagram and mark the given information. hen mark the information that you can deduce. In this case, and are supplementary to congruent angles, so. lso, by the eflexive roperty of ongruence (heorem.1). Mark given information. Mark deduced information. 1 wo angle pairs and a non-included side are congruent, so by the ongruence heorem (heorem 5.11),. ecause corresponding parts of congruent triangles are congruent,. Monitoring rogress 1. xplain how you can prove that. Help in nglish and panish at igideasmath.com 78 hapter 5 ongruent riangles

53 Using ongruent riangles for Measurement MING N O OM When you cannot easily measure a length directly, you can make conclusions about the length indirectly, usually by calculations based on known lengths. Use the following method to find the distance across a river, from point N to point. lace a stake at on the near side so that N N. ind M, the midpoint of N. ocate the point so that N and,, and M are collinear. xplain how this plan allows you to find the distance. N M OUION ecause N N and N, N and are congruent right angles. ecause M is the midpoint of N, NM M. he vertical angles M and NM are congruent. o, M MN by the ongruence heorem (heorem 5.10). hen because corresponding parts of congruent triangles are congruent, N. o, you can find the distance N across the river by measuring. N M lanning a roof Involving airs of riangles Use the given information to write a plan for proof. Given 1, 3 4 rove OUION In and, you know that 1 and. If you can show that, then you can use the ongruence heorem (heorem 5.5). o prove that, you can first prove that. You are given 1 and 3 4. by the eflexive roperty of ongruence (heorem.1). You can use the ongruence heorem (heorem 5.10) to prove that. lan for roof Use the ongruence heorem (heorem 5.10) to prove that. hen state that. Use the ongruence heorem (heorem 5.5) to prove that. Monitoring rogress Help in nglish and panish at igideasmath.com. In xample, does it matter how far from point N you place a stake at point? xplain. 3. Write a plan to prove that U U. U ection 5.7 Using ongruent riangles 79

54 roving onstructions ecall that you can use a compass and a straightedge to copy an angle. he construction is shown below. You can use congruent triangles to prove that this construction is valid. tep 1 tep tep 3 raw a segment and arcs o copy, draw a segment with initial point. raw an arc with center. Using the same radius, draw an arc with center. abel points,, and. raw an arc raw an arc with radius and center. abel the intersection. raw a ray raw. In xample 4, you will prove that. roving a onstruction Write a proof to verify that the construction for copying an angle is valid. OUION dd and to the diagram. In the construction, one compass setting determines,,, and, and another compass setting determines and. o, you can assume the following as given statements. Given,, rove lan for roof lan in ction how that, so you can conclude that the corresponding parts and are congruent. MN 1.,, ON 1. Given.. ongruence heorem (heorem 5.8) orresponding parts of congruent triangles are congruent. Monitoring rogress Help in nglish and panish at igideasmath.com 4. Use the construction of an angle bisector on page 4. What segments can you assume are congruent? 80 hapter 5 ongruent riangles

55 5.7 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. OM H NN parts of congruent triangles are congruent.. WIING escribe a situation in which you might choose to use indirect measurement with congruent triangles to find a measure rather than measuring directly. Monitoring rogress and Modeling with Mathematics In xercises 3 8, explain how to prove that the statement is true. (ee xample 1.) In xercises 13 and 14, write a proof to verify that the construction is valid. (ee xample 4.) J J 5. JM M 6. M 7. G HJ 8. W V G N M H V U W 13. ine perpendicular to a line through a point not on the line lan for roof how that by the ongruence heorem (heorem 5.8). hen show that M M using the ongruence heorem (heorem 5.5). Use corresponding parts of congruent triangles to show that M and M are right angles. 14. ine perpendicular to a line through a point on the line M In xercises 9 1, write a plan to prove that 1. (ee xample 3.) J G H 1 1 lan for roof how that by the ongruence heorem (heorem 5.8). Use corresponding parts of congruent triangles to show that and are right angles. In xercises 15 and 16, use the information given in the diagram to write a proof. 15. rove HN H G J M N ection 5.7 Using ongruent riangles 81

56 16. rove UX Y W X U Y V 0. HOUGH OVOING he ermuda riangle is a region in the tlantic Ocean in which many ships and planes have mysteriously disappeared. he vertices are Miami, an Juan, and ermuda. Use the Internet or some other resource to find the side lengths, the perimeter, and the area of this triangle (in miles). hen create a congruent triangle on land using cities as vertices. ermuda 17. MOING WIH MHMI xplain how to find the distance across the canyon. (ee xample.) Miami, an Juan, uerto ico 18. HOW O YOU I? Use the tangram puzzle. a. Which triangle(s) have an area that is twice the area of the purple triangle? b. How many times greater is the area of the orange triangle than the area of the purple triangle? 19. OO rove that the green triangles in the Jamaican flag are congruent if and is the midpoint of. 1. MING N GUMN Your friend claims that WZY can be proven congruent to YXW using the H ongruence W heorem (hm. 5.9). Is your friend correct? xplain your reasoning. Z. II HINING etermine whether each conditional statement is true or false. If the statement is false, rewrite it as a true statement using the converse, inverse, or contrapositive. a. If two triangles have the same perimeter, then they are congruent. b. If two triangles are congruent, then they have the same area. 3. NING O IION Which triangles are congruent to? elect all that apply. M X Y J Maintaining Mathematical roficiency ind the perimeter of the polygon with the given vertices. (ection 1.4) 4. ( 1, 1), (4, 1), (4, ), ( 1, ) 5. J( 5, 3), (, 1), (3, 4) G eviewing what you learned in previous grades and lessons H N 8 hapter 5 ongruent riangles

57 5.8 oordinate roofs ssential uestion How can you use a coordinate plane to write a proof? Writing a oordinate roof Work with a partner. a. Use dynamic geometry software to draw with endpoints (0, 0) and (6, 0). b. raw the vertical line x = 3. c. raw so that lies on the line x = 3. d. Use your drawing to prove that is an isosceles triangle ample oints (0, 0) (6, 0) (3, y) egments = 6 ine x = 3 Writing a oordinate roof Work with a partner. a. Use dynamic geometry software to draw with endpoints (0, 0) and (6, 0). b. raw the vertical line x = 3. c. lot the point (3, 3) and draw. hen use your drawing to prove that is an isosceles right triangle. IIUING H ONING O OH o be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results ample oints (0, 0) (6, 0) (3, 3) egments = 6 = 4.4 = 4.4 ine x = 3 d. hange the coordinates of so that lies below the x-axis and is an isosceles right triangle. e. Write a coordinate proof to show that if lies on the line x = 3 and is an isosceles right triangle, then must be the point (3, 3) or the point found in part (d). ommunicate Your nswer 3. How can you use a coordinate plane to write a proof? 4. Write a coordinate proof to prove that with vertices (0, 0), (6, 0), and (3, 3 3 ) is an equilateral triangle. ection 5.8 oordinate roofs 83

58 5.8 esson ore Vocabulary coordinate proof, p. 84 What You Will earn lace figures in a coordinate plane. Write coordinate proofs. lacing igures in a oordinate lane coordinate proof involves placing geometric figures in a coordinate plane. When you use variables to represent the coordinates of a figure in a coordinate proof, the results are true for all figures of that type. lacing a igure in a oordinate lane lace each figure in a coordinate plane in a way that is convenient for finding side lengths. ssign coordinates to each vertex. a. a rectangle b. a scalene triangle OUION It is easy to find lengths of horizontal and vertical segments and distances from (0, 0), so place one vertex at the origin and one or more sides on an axis. a. et h represent the length and b. Notice that you need to use k represent the width. three different variables. (0, k) y (h, k) y (f, g) k (0, 0) h (h, 0) x (0, 0) (d, 0) x Monitoring rogress Help in nglish and panish at igideasmath.com 1. how another way to place the rectangle in xample 1 part (a) that is convenient for finding side lengths. ssign new coordinates.. square has vertices (0, 0), (m, 0), and (0, m). ind the fourth vertex. Once a figure is placed in a coordinate plane, you may be able to prove statements about the figure. Writing a lan for a oordinate roof Write a plan to prove that O bisects. Given oordinates of vertices of O and O rove O bisects. y (0, 4) (3, 0) OUION 84 hapter 5 ongruent riangles ( 3, 0) O(0, 0) lan for roof Use the istance ormula to find the side lengths of O and O. hen use the ongruence heorem (heorem 5.8) to show that O O. inally, use the fact that corresponding parts of congruent triangles are congruent to conclude that O O, which implies that O bisects. 4 x

59 Monitoring rogress 3. Write a plan for the proof. Given GJ bisects OGH. rove GJO GJH Help in nglish and panish at igideasmath.com y 6 G 4 J O 4 H x he coordinate proof in xample applies to a specific triangle. When you want to prove a statement about a more general set of figures, it is helpful to use variables as coordinates. or instance, you can use variable coordinates to duplicate the proof in xample. Once this is done, you can conclude that O bisects for any triangle whose coordinates fit the given pattern. ( h, 0) y (0, k) (h, 0) O(0, 0) x pplying Variable oordinates lace an isosceles right triangle in a coordinate plane. hen find the length of the hypotenuse and the coordinates of its midpoint M. INING N NY OIN nother way to solve xample 3 is to place a triangle with point at (0, h) on the y-axis and hypotenuse on the x-axis. o make a right angle, position and so that legs and have slopes of 1 and 1, respectively. lope is 1. y lope is 1. (0, h) ( h, 0) (h, 0) x ength of hypotenuse = h M ( h + h, ) = M(0, 0) OUION lace O with the right angle at the origin. et the length of the legs be k. hen the vertices are located at (0, k), (k, 0), and O(0, 0). (0, k) y M O(0, 0) (k, 0) x Use the istance ormula to find, the length of the hypotenuse. = (k 0) + (0 k) = k + ( k) = k + k = k = k Use the Midpoint ormula to find the midpoint M of the hypotenuse. M ( 0 + k, k + 0 ) ( = M k, k ) o, the length of the hypotenuse is k and the midpoint of the hypotenuse is ( k, k ). Monitoring rogress Help in nglish and panish at igideasmath.com 4. Graph the points O(0, 0), H(m, n), and J(m, 0). Is OHJ a right triangle? ind the side lengths and the coordinates of the midpoint of each side. ection 5.8 oordinate roofs 85

60 Writing oordinate roofs Writing a oordinate roof Write a coordinate proof. Given oordinates of vertices of quadrilateral OUV rove OU UVO y (m, k) U(m + h, k) OUION egments OV and U have the same length. OV = h 0 = h U = (m + h) m = h O(0, 0) V(h, 0) Horizontal segments U and OV each have a slope of 0, which implies that they are parallel. egment OU intersects U and OV to form congruent alternate interior angles, UO and VOU. y the eflexive roperty of ongruence (heorem.1), OU OU. o, you can apply the ongruence heorem (heorem 5.5) to conclude that OU UVO. Writing a oordinate roof x You buy a tall, three-legged plant stand. When you place a plant on the stand, the stand appears to be unstable under the weight of the plant. he diagram at the right shows a coordinate plane superimposed on one pair of the plant stand s legs. he legs are extended to form O. rove that O is a scalene triangle. xplain why the plant stand may be unstable. y (1, 48) OUION irst, find the side lengths of O. O = (48 0) + (1 0) = = (18 1) + (0 48) = O = 18 0 = 18 O(0, 0) 1 (18, 0) x ecause O has no congruent sides, O is a scalene triangle by definition. he plant stand may be unstable because O is longer than, so the plant stand is leaning to the right. Monitoring rogress 5. Write a coordinate proof. Given oordinates of vertices of NO and NMO rove NO NMO Help in nglish and panish at igideasmath.com y (0, h) N(h, h) O(0, 0) M(h, 0) x 86 hapter 5 ongruent riangles

61 5.8 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. VOUY How is a coordinate proof different from other types of proofs you have studied? How is it the same?. WIING xplain why it is convenient to place a right triangle on the grid as shown when writing a coordinate proof. (0, b) y (0, 0) (a, 0) x Monitoring rogress and Modeling with Mathematics In xercises 3 6, place the figure in a coordinate plane in a convenient way. ssign coordinates to each vertex. xplain the advantages of your placement. (ee xample 1.) 3. a right triangle with leg lengths of 3 units and units 4. a square with a side length of 3 units 5. an isosceles right triangle with leg length p 6. a scalene triangle with one side length of m In xercises 7 and 8, write a plan for the proof. (ee xample.) 7. Given oordinates of vertices of OM and ONM rove OM and ONM are isosceles triangles. 4 y (3, 4) M(8, 4) In xercises 9 1, place the figure in a coordinate plane and find the indicated length. 9. a right triangle with leg lengths of 7 and 9 units; ind the length of the hypotenuse. 10. an isosceles triangle with a base length of 60 units and a height of 50 units; ind the length of one of the legs. 11. a rectangle with a length of 5 units and a width of 4 units; ind the length of the diagonal. 1. a square with side length n; ind the length of the diagonal. In xercises 13 and 14, graph the triangle with the given vertices. ind the length and the slope of each side of the triangle. hen find the coordinates of the midpoint of each side. Is the triangle a right triangle? isosceles? xplain. (ssume all variables are positive and m n.) (ee xample 3.) 13. (0, 0), (h, h), (h, 0) O(0, 0) N(5, 0) 8. Given G is the midpoint of H. rove GHJ GO 8 x 14. (0, n), (m, n), (m, 0) In xercises 15 and 16, find the coordinates of any unlabeled vertices. hen find the indicated length(s). 15. ind ON and MN. 16. ind O. y 4 H(1, 4) J(6, 4) G O(0, 0) 4 (5, 0) x O(0, 0) y N (h, 0) k units M(h, 0) x O y U k units x ection 5.8 oordinate roofs 87

62 OO In xercises 17 and 18, write a coordinate proof. (ee xample 4.) 17. Given oordinates of vertices of and O rove O y (h, k) (h, k) O(0, 0) (h, 0) (h, k) 18. Given oordinates of, H is the midpoint of, G is the midpoint of. rove G H H y (0, k) G x. ONING he vertices of a parallelogram are (w, 0), (0, v), ( w, 0), and (0, v). What is the midpoint of the side in uadrant III? ( w, v ) ( w ) ), v ( w, v ) ( w, v 3. ONING rectangle with a length of 3h and a width of k has a vertex at ( h, k). Which point cannot be a vertex of the rectangle? (h, k) ( h, 0) (h, 0) (h, k) 4. HOUGH OVOING hoose one of the theorems you have encountered up to this point that you think would be easier to prove with a coordinate proof than with another type of proof. xplain your reasoning. hen write a coordinate proof. ( h, 0) O (h, 0) 19. MOING WIH MHMI You and your cousin are camping in the woods. You hike to a point that is 500 meters east and 100 meters north of the campsite. Your cousin hikes to a point that is 1000 meters east of the campsite. Use a coordinate proof to prove that the triangle formed by your position, your cousin s position, and the campsite is isosceles. (ee xample 5.) x 5. II HINING he coordinates of a triangle are (5d, 5d ), (0, 5d ), and (5d, 0). How should the coordinates be changed to make a coordinate proof easier to complete? 6. HOW O YOU I? Without performing any calculations, how do you know that the diagonals of square UVW are perpendicular to each other? How can you use a similar diagram to show that the diagonals of any square are perpendicular to each other? W(, 0) 4 y 3 (0, ) U(, 0) 4 x 0. MING N GUMN wo friends see a drawing of quadrilateral with vertices (0, ), (3, 4), (1, 5), and (, 1). One friend says the quadrilateral is a parallelogram but not a rectangle. he other friend says the quadrilateral is a rectangle. Which friend is correct? Use a coordinate proof to support your answer. 1. MHMI ONNION Write an algebraic expression for the coordinates of each endpoint of a line segment whose midpoint is the origin. Maintaining Mathematical roficiency V(0, ) 3 7. OO Write a coordinate proof for each statement. a. he midpoint of the hypotenuse of a right triangle is the same distance from each vertex of the triangle. b. ny two congruent right isosceles triangles can be combined to form a single isosceles triangle. YW bisects XYZ such that m XYW = (3x 7) and m WYZ = (x + 1). (ection 1.5) 8. ind the value of x. 9. ind m XYZ. eviewing what you learned in previous grades and lessons 88 hapter 5 ongruent riangles

63 What id You earn? ore Vocabulary legs (of a right triangle), p. 64 hypotenuse (of a right triangle), p. 64 coordinate proof, p. 84 ore oncepts heorem 5.8 ide-ide-ide () ongruence heorem, p. 6 heorem 5.9 Hypotenuse-eg (H) ongruence heorem, p. 64 heorem 5.10 ngle-ide-ngle () ongruence heorem, p. 70 heorem 5.11 ngle-ngle-ide () ongruence heorem, p. 71 Using ongruent riangles, p. 78 roving onstructions, p. 80 lacing igures in a oordinate lane, p. 84 Writing oordinate roofs, p. 86 Mathematical ractices 1. Write a simpler problem that is similar to xercise on page 67. escribe how to use the simpler problem to gain insight into the solution of the more complicated problem in xercise.. Make a conjecture about the meaning of your solutions to xercises 1 3 on page Identify at least two external resources that you could use to help you solve xercise 0 on page 8. ongruent triangles are often used to create company logos. Why are they used and what are the properties that make them attractive? ollowing the required constraints, create your new logo and justify how your shape contains the required properties. o explore the answers to these questions and more, go to igideasmath.com. erformance ask reating the ogo 89

64 5 hapter eview 5.1 ngles of riangles (pp ) ynamic olutions available at igideasmath.com lassify the triangle by its sides and by measuring its angles. he triangle does not have any congruent sides, so it is scalene. he measure of is 117, so the triangle is obtuse. he triangle is an obtuse scalene triangle. 1. lassify the triangle at the right by its sides and by measuring its angles. ind the measure of the exterior angle (9x + 9) 46 x 45 5x ind the measure of each acute angle (7x + 6) 8x 7x (6x 7) 5. ongruent olygons (pp ) Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. he diagram indicates that. orresponding angles,, orresponding sides,, 6. In the diagram, GHJ MN. Identify all pairs of congruent corresponding parts. hen write another congruence statement for the quadrilaterals. G H M N 7. ind m V. U 74 J V 90 hapter 5 ongruent riangles

65 5.3 roving riangle ongruence by (pp ) Write a proof. Given, rove MN 1.. ON 1. Given. Given Vertical ngles ongruence heorem (heorem.6) ongruence heorem (heorem 5.5) ecide whether enough information is given to prove that WXZ YZX using the ongruence heorem (heorem 5.5). If so, write a proof. If not, explain why. 8. W X 9. W X Z Y Z Y 5.4 quilateral and Isosceles riangles (pp ) In MN, M N. Name two congruent angles. M N, so by the ase ngles heorem (heorem 5.6), M N. opy and complete the statement. 10. If, then. M N 11. If V V, then. 1. If, then. 13. If V V, then. V 14. ind the values of x and y in the diagram. 6 8x 5y + 1 hapter 5 hapter eview 91

66 5.5 roving riangle ongruence by (pp ) Write a proof. Given, rove MN 1.. ON 1. Given. Given eflexive roperty of ongruence (heorem.1) ongruence heorem (heorem 5.8) 15. ecide whether enough information is given to prove that M NM using the ongruence heorem (hm. 5.8). If so, write a proof. If not, explain why. M N 16. ecide whether enough information is given to prove that WXZ YZX using the H ongruence heorem (hm. 5.9). If so, write a proof. If not, explain why. W X Z Y 5.6 roving riangle ongruence by and (pp ) Write a proof. Given, rove MN 1. ON 1. Given.. Given Vertical ngles ongruence heorem (hm..6) ongruence heorem (hm. 5.11) 9 hapter 5 ongruent riangles

67 ecide whether enough information is given to prove that the triangles are congruent using the ongruence heorem (hm. 5.11). If so, write a proof. If not, explain why. 17. G, HJ 18. UV, H V G J U ecide whether enough information is given to prove that the triangles are congruent using the ongruence heorem (hm. 5.10). If so, write a proof. If not, explain why. 19. N, MN 0. WXZ, YZX W X N M Z Y 5.7 Using ongruent riangles (pp. 77 8) xplain how you can prove that. If you can show that, then you will know that. You are given and. You know that by the eflexive roperty of ongruence (hm..1). wo pairs of sides and their included angles are congruent, so by the ongruence heorem (hm. 5.5),. ecause corresponding parts of congruent triangles are congruent,. 1. xplain how to prove that N. M H J N. Write a plan to prove that 1. 1 V U W hapter 5 hapter eview 93

68 5.8 oordinate roofs (pp ) Write a coordinate proof. Given oordinates of vertices of O and y rove O (j, j) (j, j) egments O and have the same length. O = ( j 0) + ( j 0) = j + j = j = j = ( j j) + ( j 0) = ( j) + j = j = j egments and have the same length. = = j = (j j) + (j j) = j + j = j = j egments O and have the same length. O = j 0 = j = j 0 = j O(0, 0) (j, 0) x o, you can apply the ongruence heorem (heorem 5.8) to conclude that O. 3. Write a coordinate proof. Given oordinates of vertices of quadrilateral O rove O O y (h, k + j) (0, j) (h, k) O(0, 0) x 4. lace an isosceles triangle in a coordinate plane in a way that is convenient for finding side lengths. ssign coordinates to each vertex. 5. rectangle has vertices (0, 0), (k, 0), and (0, k). ind the fourth vertex. 94 hapter 5 ongruent riangles

69 5 hapter est Write a proof. 1. Given. Given J M, MJ 3. Given, rove rove MJ M rove J N M 4. ind the measure of each acute angle in the figure at the right. 5. Is it possible to draw an equilateral triangle that is not equiangular? If so, provide an example. If not, explain why. 6. an you use the hird ngles heorem (heorem 5.4) to prove that two triangles are congruent? xplain your reasoning. (4x ) (3x + 8) Write a plan to prove that V Z W X Y 9. Is there more than one theorem that could be used to prove that? If so, list all possible theorems. y Write a coordinate proof to show that the triangles created by the keyboard stand are congruent x he picture shows the yramid of estius, which is located in ome, Italy. he measure of the base for the triangle shown is 100 oman feet. he measures of the other two sides of the triangle are both 144 oman feet. a. lassify the triangle shown by its sides. b. he measure of 3 is 40. What are the measures of 1 and? xplain your reasoning. 1 hapter 5 hapter est 95