Geometry. Chapter 1 Resource Masters

Transcription

1 Geometry hapter esource Masters

2 eading to Learn Mathematics Vocabulary uilder This is an alphabetical list of the key vocabulary terms you will learn in hapter. s you study the chapter, complete each term s definition or description. emember to add the page number where you found the term. dd these pages to your Geometry tudy otebook to review vocabulary at the end of the chapter. acute angle Vocabulary Term ound on age efinition/escription/xample Vocabulary uilder adjacent angles uh JY suhnt angle angle bisector collinear koh LI ee uhr complementary angles congruent kuhn GOO uhnt coplanar koh LY nuhr line segment linear pair (continued on the next page) Glencoe/McGraw-Hill vii Glencoe Geometry

3 eading to Learn Mathematics Vocabulary uilder (continued) midpoint Vocabulary Term ound on age efinition/escription/xample obtuse angle perimeter perpendicular lines polygon HL ee gahn ray right angle segment bisector supplementary angles vertical angles Glencoe/McGraw-Hill viii Glencoe Geometry

4 - tudy Guide and Intervention oints, Lines, and lanes ame oints, Lines, and lanes In geometry, a point is a location, a line contains points, and a plane is a flat surface that contains points and lines. If points are on the same line, they are collinear. If points on are the same plane, they are coplanar. xample a. a line containing point Use the figure to name each of the following. The line can be named as. lso, any two of the three points on the line can be used to name it.,, or b. a plane containing point The plane can be named as plane or can be named using three noncollinear points in the plane, such as plane, plane, and so on. Lesson - xercises efer to the figure.. ame a line that contains point.. What is another name for line m? m 3. ame a point not on. 4. ame the intersection of and. 5. ame a point not on line or line m. raw and label a plane for each relationship. 6. is in plane. 7. T intersects at. T X Y 8. oint X is collinear with points and. 9. oint Y is not collinear with points T and. 0. Line contains points X and Y. Glencoe/McGraw-Hill Glencoe Geometry

5 - tudy Guide and Intervention (continued) oints, Lines, and lanes oints, Lines, and lanes in pace pace is a boundless, three-dimensional set of all points. It contains lines and planes. xample a. How many planes appear in the figure? O There are three planes: plane, plane O, and plane. b. re points,, and coplanar? Yes. They are contained in plane O. xercises efer to the figure.. ame a line that is not contained in plane.. ame a plane that contains point. 3. ame three collinear points. efer to the figure. 4. How many planes are shown in the figure? 5. re points,, G, and H coplanar? xplain. G J H I 6. ame a point coplanar with,, and. raw and label a figure for each relationship. 7. lanes M and intersect in HJ. t M s 8. Line r is in plane, line s is in plane M, and lines r and s intersect at point J. H J r 9. Line t contains point H and line t does not lie in plane M or plane. Glencoe/McGraw-Hill Glencoe Geometry

6 - kills ractice efer to the figure. oints, Lines, and lanes. ame a line that contains point.. ame a point contained in line n. p n G 3. What is another name for line p? 4. ame the plane containing lines n and p. Lesson - raw and label a figure for each relationship. 5. oint K lies on T. 6. lane J contains line s. K T J s 7. Y lies in plane and contains point, but does not contain point H. in plane U. Y H 8. Lines q and f intersect at point Z U q Z f efer to the figure. 9. How many planes are shown in the figure? 0. How many of the planes contain points and? W. ame four points that are coplanar.. re points,, and coplanar? xplain. Glencoe/McGraw-Hill 3 Glencoe Geometry

7 - ractice efer to the figure. oints, Lines, and lanes. ame a line that contains points T and.. ame a line that intersects the plane containing points,, and. j M T g h 3. ame the plane that contains T and. raw and label a figure for each relationship. 4. K and G intersect at point M in plane T. T M K G 5. line contains L( 4, 4) and M(, 3). Line q is in the same coordinate plane but does not intersect LM. Line q contains point. y M q O x L efer to the figure. T 6. How many planes are shown in the figure? 7. ame three collinear points. W X 8. re points,,, and W coplanar? xplain. M VIULIZTIO ame the geometric term(s) modeled by each object tip of pin. TO strings. a car antenna 3. a library card Glencoe/McGraw-Hill 4 Glencoe Geometry

8 - eading to Learn Mathematics oints, Lines, and lanes re-ctivity Why do chairs sometimes wobble? eading the Lesson. omplete each sentence. ead the introduction to Lesson - at the top of page 6 in your textbook. ind three pencils of different lengths and hold them upright on your desk so that the three pencil points do not lie along a single line. an you place a flat sheet of paper or cardboard so that it touches all three pencil points? How many ways can you do this if you keep the pencil points in the same position? How will your answer change if there are four pencil points? Lesson - a. oints that lie on the same lie are called points. b. oints that do not lie in the same plane are called points. c. There is exactly one through any two points. d. There is exactly one through any three noncollinear points.. efer to the figure at the right. Indicate whether each statement is true or false. a. oints,, and are collinear. b. The intersection of plane and line m is point. U c. Line and line m do not intersect. m d. oints,,and can be used to name plane U. e. Line lies in plane. 3. omplete the figure at the right to show the following relationship: Lines, m, and n are coplanar and lie in plane. Lines and m intersect at point. Line n intersects line m at, but does not intersect line. m n Helping You emember 4. ecall or look in a dictionary to find the meaning of the prefix co-.what does this prefix mean? How can it help you remember the meaning of collinear? Glencoe/McGraw-Hill 5 Glencoe Geometry

9 - nrichment oints and Lines on a Matrix matrix is a rectangular array of rows and columns. oints and lines on a matrix are not defined in the same way as in uclidean geometry. point on a matrix is a dot, which can be small or large. line on a matrix is a path of dots that line up. etween two points on a line there may or may not be other points. Three examples of lines are shown at the upper right. The broad line can be thought of as a single line or as two narrow lines side by side. ot-matrix printers for computers used dots to form characters. The dots are often called pixels. The matrix at the right shows how a dot-matrix printer might print the letter. raw points on each matrix to create the given figures.. raw two intersecting lines that have. raw two lines that cross but have four points in common. no common points. 3. Make the number 0 (zero) so that it 4. Make the capital letter O so that it extends to the top and bottom sides extends to each side of the matrix. of the matrix. 5. Using separate grid paper, make dot designs for several other letters. Which were the easiest and which were the most difficult? Glencoe/McGraw-Hill 6 Glencoe Geometry

10 - tudy Guide and Intervention Linear Measure and recision Measure Line egments part of a line between two endpoints is called a line segment. The lengths of M and are written as M and.when you measure a segment, the precision of the measurement is half of the smallest unit on the ruler. xample xample ind the length of M. M ind the length of. cm 3 4 in. The long marks are centimeters, and the shorter marks are millimeters. The length of M is 3.4 centimeters. The measurement is accurate to within 0.5 millimeter, so M is between 3.35 centimeters and 3.45 centimeters long. xercises The long marks are inches and the short marks are quarter inches. The length of is about 3 4 inches. The measurement is accurate to within one half of a quarter inch, or 8 inch, so is between 5 inches and ind the length of each line segment or object. inches long. Lesson -.. T cm 3 in in. cm 3 ind the precision for each measurement in mm cm 8. ft mm 0. yd Glencoe/McGraw-Hill 7 Glencoe Geometry

11 - tudy Guide and Intervention (continued) Linear Measure and recision alculate Measures On, to say that point M is between points and means,, and M are collinear and M M. On, 3 cm. We can say that the segments are congruent, or. lashes on the figure indicate which segments are congruent. M xample xample ind. ind x and.. cm.9 cm alculate by adding and Therefore, is 3. centimeters long. x 5 x x is between and. x x x 5 3x x 5 x 5 x 5 (5) 5 5 xercises ind the measurement of each segment. ssume that the art is not drawn to scale.. T..0 cm.5 cm T 3 4 in. 6 in. 3. X Z 3 4. W X 3 in. 4 in. X Y Z W 6 cm X Y ind x and if is between and T. 5. 5x, T 3x, and T x, T 5x 4, and T x, T, and T x, T, and T 4. Use the figures to determine whether each pair of segments is congruent. 9. and cm 0. X Y and Y Z X 5 cm 5 cm cm 3x 5 5x Y 9x Z Glencoe/McGraw-Hill 8 Glencoe Geometry

12 - kills ractice Linear Measure and recision ind the length of each line segment or object... cm in. ind the precision for each measurement feet 4. centimeters 5. 9 inches ind the measurement of each segment G H in. 4 in. 4.9 cm 5. cm 9.7 mm G H 5 mm Lesson - ind the value of the variable and YZ if Y is between X and Z. 9. XY 5p, YZ p, and XY 5 0. XY, YZ g, and XZ 8. XY 4m, YZ 3m, and XZ 4. XY c, YZ 6c, and XZ 8 Use the figures to determine whether each pair of segments is congruent. 3., 4. M, 5. W X, W Z m 9 ft yd Y Z 3 m 3 m M 0 yd 5 ft 5 ft 5 m yd X W Glencoe/McGraw-Hill 9 Glencoe Geometry

13 - ractice Linear Measure and recision ind the length of each line segment or object... in. cm ind the precision for each measurement meters 4. 7 inches millimeters 4 ind the measurement of each segment W X 8.4 cm 4.7 cm 3 8 in. 4 in. W X 89.6 cm Y 00 cm ind the value of the variable and KL if K is between J and L. 9. JK 6r, KL 3r, and JL 7 0. JK s, KL s, and JL 5s 0 Use the figures to determine whether each pair of segments is congruent.. T U, W., 3. G, T ft.7 in. 5x G H ft 3 ft 6x U W 3 ft.9 in. 4. TY Jorge used the figure at the right to make a pattern for a mosaic he plans to inlay on a tabletop. ame all of the congruent segments in the figure. Glencoe/McGraw-Hill 0 Glencoe Geometry

14 - eading to Learn Mathematics Linear Measure and recision re-ctivity Why are units of measure important? ead the introduction to Lesson - at the top of page 3 in your textbook. The basic unit of length in the metric system is the meter. How many meters are there in one kilometer? o you think it would be easier to learn the relationships between the different units of length in the customary system (used in the United tates) or in the metric system? xplain your answer. eading the Lesson. xplain the difference between a line and a line segment and why one of these can be measured, while the other cannot.. What is the smallest length marked on a -inch ruler? What is the smallest length marked on a centimeter ruler? 3. ind the precision of each measurement. a. 5 cm b. 5.0 cm Lesson - 4. efer to the figure at the right. Which one of the following statements is true? xplain your answer. 4.5 cm 4.5 cm 5. uppose that is a point on V W and is not the same point as V or W.Tell whether each of the following statements is always, sometimes, or never true. a. V W b. is between V and W. c. V VW W Helping You emember 6. good way to remember terms used in mathematics is to relate them to everyday words you know. Give three words that are used outside of mathematics that can help you remember that there are 00 centimeters in a meter. Glencoe/McGraw-Hill Glencoe Geometry

15 - nrichment oints quidistant from egments The distance from a point to a segment is zero if the point is on the segment. Otherwise, it is the length of the shortest segment from the point to the segment. figure is a locus if it is the set of all points that satisfy a set of conditions. The locus of all points that are 4 inch from the segment is shown by two dashed segments with semicircles at both ends.. uppose,,, and are four different points, and consider the locus of all points x units from and x units from. Use any unit you find convenient. The locus can take different forms. ketch at least three possibilities. List some of the things that seem to affect the form of the locus. X Y X Y. onduct your own investigation of the locus of points equidistant from two segments. escribe your results on a separate sheet of paper. Glencoe/McGraw-Hill Glencoe Geometry

16 -3 tudy Guide and Intervention istance and Midpoints istance etween Two oints istance on a umber Line istance in the oordinate lane a b b a or a b ythagorean Theorem: a b c istance ormula: d (x x ) (y y ) (, ) O y (, 3) x (, ) xample xample ind ( 4) 6 6 ind the distance between (, ) and (, 3). ythagorean Theorem () () () () (3) (4) () istance ormula d (x x ) (y y ) ( )) ( (3 ( )) (3) (4) 5 5 xercises Use the number line to find each measure... G G 6. G G Lesson -3 Use the ythagorean Theorem to find the distance between each pair of points. 9. (0, 0), (6, 8) 0. (, 3), (3, 5). M(, ), (9, 3). (, ), ( 9, 6) Use the istance ormula to find the distance between each pair of points. 3. (0, 0), (5, 0) 4. O(, 0), ( 8, 3) 5. (, ), (6, ) 6. (, 0), ( 4, 3) Glencoe/McGraw-Hill 3 Glencoe Geometry

17 -3 tudy Guide and Intervention (continued) istance and Midpoints Midpoint of a egment Midpoint on a If the coordinates of the endpoints of a segment are a and b, umber Line then the coordinate of the midpoint of the segment is a. b If a segment has endpoints with coordinates (x Midpoint on a, y ) and (x, y ), oordinate lane then the coordinates of the midpoint of the segment are x x, y y. xample ind the coordinate of the midpoint of. 3 0 The coordinates of and are 3 and. If M is the midpoint of, then the coordinate of M is 3 or. xample M is the midpoint of for (, 4) and (4, ). ind the coordinates of M. x M x, y y 4, 4 or (,.5) xercises Use the number line to find the coordinate of the midpoint of each segment... G G G G ind the coordinates of the midpoint of a segment having the given endpoints. 9. (0, 0), (, 8) 0. (, 8), (6, ). M(, ), ( 9, 3). (, 6), ( 9, 3) 3. (0, ), T(9, 0) 4. M(, ), ( 9, 6) Glencoe/McGraw-Hill 4 Glencoe Geometry

18 -3 kills ractice istance and Midpoints Use the number line to find each measure. J K L M. L. JL K 4. M Use the ythagorean Theorem to find the distance between each pair of points y y G O x O x 7. K(, 3), (4, 4) 8. ( 3, ), (, 3) Use the istance ormula to find the distance between each pair of points. 9. Y(, 0), (, 6) 0. W(, ), (5, ). ( 7, 3), (5, ). ( 3, ), (, 6) Use the number line to find the coordinate of the midpoint of each segment Lesson ind the coordinates of the midpoint of a segment having the given endpoints. 7. T(3, ), U(5, 3) 8. J( 4, ), (5, ) ind the coordinates of the missing endpoint given that is the midpoint of. 9. (, 0), (5, ) 0. (5, 4), (6, 3). (3, 9), (, 5) Glencoe/McGraw-Hill 5 Glencoe Geometry

19 -3 ractice istance and Midpoints Use the number line to find each measure.. VW. TV 3. T 4. V T U V W Use the ythagorean Theorem to find the distance between each pair of points. 5. y 6. Z y M O x O x Use the istance ormula to find the distance between each pair of points. 7. L( 7, 0), Y(5, 9) 8. U(, 3), (4, 6) Use the number line to find the coordinate of the midpoint of each segment. 9. T 0. T T. ind the coordinates of the midpoint of a segment having the given endpoints. 3. K( 9, 3), H(5, 7) 4. W(, 7), T( 8, 4) ind the coordinates of the missing endpoint given that is the midpoint of. 5. (5, 8), (4, 3) 6. (, 9), (, 6) 7. ( 3, 8), (, ) 8. IMT The coordinates of the vertices of a quadrilateral are (, 3), (3, 3), T(5, ), and U(, ). ind the perimeter of the quadrilateral. ound to the nearest tenth. Glencoe/McGraw-Hill 6 Glencoe Geometry

20 -3 eading to Learn Mathematics istance and Midpoints re-ctivity How can you find the distance between two points without a ruler? eading the Lesson ead the introduction to Lesson -3 at the top of page in your textbook. Look at the triangle in the introduction to this lesson. What is the special name for in this triangle? ind in this figure. Write your answer both as a radical and as a decimal number rounded to the nearest tenth.. Match each formula or expression in the first column with one of the names in the second column. a. d (x x ) ( y y ) b. a b i. ythagorean Theorem ii. istance ormula in the oordinate lane c. XY a b iii. Midpoint of a egment in the oordinate lane d. c a b iv. istance ormula on a umber Line x y e. x, y v. Midpoint of a egment on a umber Line. ill in the steps to calculate the distance between the points M(4, 3) and (, 7). Let (x, y ) (4, 3). Then (x, y ) (, ). d ( ) ( ) M ( ) ( ) M ( ) ( ) M M ind a decimal approximation for M to the nearest hundredth. Lesson -3 Helping You emember 3. good way to remember a new formula in mathematics is to relate it to one you already know. If you forget the istance ormula, how can you use the ythagorean Theorem to find the distance d between two points on a coordinate plane? Glencoe/McGraw-Hill 7 Glencoe Geometry

21 -3 nrichment Lengths on a Grid venly-spaced horizontal and vertical lines form a grid. You can easily find segment lengths on a grid if the endpoints are grid-line intersections. or horizontal or vertical segments, simply count squares. or diagonal segments, use the ythagorean Theorem (proven in hapter 7). This theorem states that in any right triangle, if the length of the longest side (the side opposite the right angle) is c and the two shorter sides have lengths a and b, then c a b. ind the measure of on the grid at the right. Locate a right triangle with as its longest side. xample units I J L K M ind each measure to the nearest tenth of a unit.. I J. M I K 6. J K 7. L M 8. L Use the grid above. ind the perimeter of each triangle to the nearest tenth of a unit LM 3. Of all the segments shown on the 4. On the grid, unit 0.5 cm. How can the grid, which is longest? What is its answers above be used to find the measures length? in centimeters? 5. Use your answer from exercise 8 to 6. Use a centimeter ruler to find the perimeter calculate the length of segment L of triangle IJK to the nearest tenth of a in centimeters. heck by measuring centimeter. with a centimeter ruler. Glencoe/McGraw-Hill 8 Glencoe Geometry

22 -4 tudy Guide and Intervention ngle Measure Measure ngles If two noncollinear rays have a common endpoint, they form an angle. The rays are the sides of the angle. The common endpoint is the vertex. The angle at the right can be named as,,, or. right angle is an angle whose measure is 90. n acute angle has measure less than 90. n obtuse angle has measure greater than 90 but less than 80. xample xample T 3 Measure each angle and classify it as right, acute, or obtuse. a. ame all angles that have as a vertex. Three angles are,, and 3. or other angles, use three letters to name them:, T, and T. b. ame the sides of., a. Using a protractor, m , so is an acute angle. b. Using a protractor, m , so is an obtuse angle. c. Using a protractor, m 90. is a right angle. xercises efer to the figure.. ame the vertex of 4.. ame the sides of. 3. Write another name for. 4 3 Lesson -4 Measure each angle in the figure and classify it as right, acute, or obtuse. 4. M M Glencoe/McGraw-Hill 9 Glencoe Geometry

23 -4 tudy Guide and Intervention (continued) ngle Measure ongruent ngles ngles that have the same measure are congruent angles. ray that divides an angle into two congruent angles is called an angle bisector. In the figure, is the angle bisector of M. oint lies in the interior of M and M. M xample efer to the figure above. If m M x 4 and m x 34, find x and find m M. ince bisects M, M, or m M m. x 4 x 34 m (x 4) (x 34) x 4 x x 34 x x x x 0 xercises bisects T, and and are opposite rays.. If m T 60 and m 4x 4, find the value of x. T. If m 3x 3 and m T 6x, find m T. and are opposite rays, bisects, and bisects. 3. If m 6x 4 and m 7x, find m If m 4x 0 and m 5x, find m. 5. If m 6y and m 8y 4, find m. 6. Is a right angle? xplain. Glencoe/McGraw-Hill 0 Glencoe Geometry

24 -4 kills ractice ngle Measure or xercises, use the figure at the right. ame the vertex of each angle W U 4 T 5 3 V ame the sides of each angle TV 8. Write another name for each angle WT. Measure each angle and classify it as right, acute, or obtuse. 3. M 4. OM O 5. M 6. MO LG In the figure, and are opposite rays, bisects, and bisects. 7. If m 4x 6 and m 6x 4, find m. L M Lesson If m 7x 8 and m 5x 0, find m. Glencoe/McGraw-Hill Glencoe Geometry

25 -4 ractice ngle Measure or xercises 0, use the figure at the right. ame the vertex of each angle M ame the sides of each angle M O MO 8. OM Write another name for each angle Measure each angle and classify it as right, acute, or obtuse. V W. UZW. YZW U X 3. TZW 4. UZT T Z Y LG In the figure, and are opposite rays, bisects, and G bisects. 5. If m 4x 5 and m 6x 5, find m. 6. If m G 9x 3 and m G 3x 9, find m G. G 7. TI IG The diagram shows a sign used to warn drivers of a school zone or crossing. Measure and classify each numbered angle. Glencoe/McGraw-Hill Glencoe Geometry

26 -4 eading to Learn Mathematics ngle Measure re-ctivity How big is a degree? ead the introduction to Lesson -4 at the top of page 9 in your textbook. semicircle is half a circle. How many degrees are there in a semicircle? How many degrees are there in a quarter circle? eading the Lesson. Match each description in the first column with one of the terms in the second column. ome terms in the second column may be used more than once or not at all. a. a figure made up of two noncollinear rays with a. vertex common endpoint. angle bisector b. angles whose degree measures are less than opposite rays c. angles that have the same measure 4. angle d. angles whose degree measures are between 90 and obtuse angles e. a tool used to measure angles 6. congruent angles f. the common endpoint of the rays that form an angle 7. right angles g. a ray that divides an angle into two congruent angles 8. acute angles 9. compass 0. protractor. Use the figure to name each of the following. a. a right angle b. an obtuse angle c. an acute angle d. a point in the interior of e. a point in the exterior of f. the angle bisector of g. a point on h. the sides of i. a pair of opposite rays j. the common vertex of all angles shown in the figure k. a pair of congruent angles l. the angle with the greatest measure Helping You emember 3. good way to remember related geometric ideas is to compare them and see how they are alike and how they are different. Give some similarities and differences between congruent segments and congruent angles. 8 8 G Lesson -4 Glencoe/McGraw-Hill 3 Glencoe Geometry

27 -4 nrichment ngle elationships ngles are measured in degrees ( ). ach degree of an angle is divided into 60 minutes ( ), and each minute of an angle is divided into 60 seconds ( ) Two angles are complementary if the sum of their measures is 90. ind the complement of each of the following angles Two angles are supplementary if the sum of their measures is 80. ind the supplement of each of the following angles Glencoe/McGraw-Hill 4 Glencoe Geometry

28 -5 tudy Guide and Intervention ngle elationships airs of ngles djacent angles are angles in the same plane that have a common vertex and a common side, but no common interior points. Vertical angles are two nonadjacent angles formed by two intersecting lines. pair of adjacent angles whose noncommon sides are opposite rays is called a linear pair. xample Identify each pair of angles as adjacent angles, vertical angles, and/or as a linear pair. a. c. T and TU have a common vertex and a common side, but no common interior points. They are adjacent angles. T U and 5 are adjacent angles whose noncommon sides are opposite rays. The angles form a linear pair. xercises b. d. M and 3 are nonadjacent angles formed by two intersecting lines. They are vertical angles. and 4 are also vertical angles G and are two angles whose measures have a sum of 90. They are complementary. and G are two angles whose measures have a sum of 80. They are supplementary. Identify each pair of angles as adjacent, vertical, and/or as a linear pair. U T. and. and 6 V and and or xercises 5 7, refer to the figure at the right. 5. Identify two obtuse vertical angles. V 6. Identify two acute adjacent angles. U T 7. Identify an angle supplementary to TU. 8. ind the measures of two complementary angles if the difference in their measures is 8. Lesson -5 Glencoe/McGraw-Hill 5 Glencoe Geometry

29 -5 tudy Guide and Intervention (continued) ngle elationships erpendicular Lines Lines, rays, and segments that form four right angles are perpendicular. The right angle symbol indicates that the lines are perpendicular. In the figure at the right, is perpendicular to, or. xample ind x so that Z Z. If Z Z, then m Z 90. m Z m Z m Z um of parts whole (9x 5) (3x ) 90 ubstitution x 6 90 implify. x 84 x 7 ivide each side by. ubtract 6 from each side. (9x 5) (3x ) Z xercises. ind x and y so that M.. ind m M. M 5x (9y 8) x 3. m 3x 0, m x, and. ind x. 4. If m 7y 3 and m 3y 3, find y so that. 5. ind x, m, and m. 3x (8x ) 6. ind y, m T, and m TW. T (4y 5) W (y 5) V Glencoe/McGraw-Hill 6 Glencoe Geometry

30 -5 kills ractice ngle elationships or xercises 6, use the figure at the right and a protractor.. ame two acute vertical angles. K. ame two obtuse vertical angles. H G J 3. ame a linear pair. 4. ame two acute adjacent angles. 5. ame an angle complementary to KH. 6. ame an angle supplementary to KG. 7. ind the measures of an angle and its complement if one angle measures 8 degrees more than the other. 8. The measure of the supplement of an angle is 36 less than the measure of the angle. ind the measures of the angles. LG or xercises 9 0, use the figure at the right. 9. If m T 8x 8, find x so that T T. 0. If m T 3y 0 and m T y, find y so that T is a right angle. T etermine whether each statement can be assumed from the figure. xplain.. WZU is a right angle. W V. YZU and UZV are supplementary. X Y Z U 3. VZU is adjacent to YZX. Lesson -5 Glencoe/McGraw-Hill 7 Glencoe Geometry

31 -5 ractice ngle elationships or xercises 4, use the figure at the right and a protractor.. ame two obtuse vertical angles.. ame a linear pair whose vertex is. 3. ame an angle not adjacent to but complementary to G. G H 4. ame an angle adjacent and supplementary to. 5. Two angles are complementary. The measure of one angle is more than twice the measure of the other angle. ind the measures of the angles. 6. If a supplement of an angle has a measure 78 less than the measure of the angle, what are the measures of the angles? LG or xercises 7 8, use the figure at the right. 7. If m G 5x 0, find x so that. 8. If m G 6x 4 and m G x 3, find x so that G is a right angle. G etermine whether each statement can be assumed from the figure. xplain. 9. O and O are complementary. M O 0. and is a linear pair.. M and M are vertical angles.. TT M arren sketched a map of the cross streets nearest to his home for his friend Miguel. escribe two different angle relationships between the streets. eacon Olive Main Glencoe/McGraw-Hill 8 Glencoe Geometry

32 -5 eading to Learn Mathematics ngle elationships re-ctivity What kinds of angles are formed when streets intersect? ead the introduction to Lesson -5 at the top of page 37 in your textbook. How many separate angles are formed if three lines intersect at a common point? (o not use an angle whose interior includes part of another angle.) How many separate angles are formed if n lines intersect at a common point? (o not count an angle whose interior includes part of another angle.) eading the Lesson. ame each of the following in the figure at the right. a. two pairs of congruent angles b. a pair of acute vertical angles c. a pair of obtuse vertical angles d. four pairs of adjacent angles e. two pairs of vertical angles f. four linear pairs g. four pairs of supplementary angles Tell whether each statement is always, sometimes, or never true. a. If two angles are adjacent angles, they form a linear pair. b. If two angles form a linear pair, they are complementary. c. If two angles are supplementary, they are congruent. d. If two angles are complementary, they are adjacent. e. When two perpendicular lines intersect, four congruent angles are formed. f. Vertical angles are supplementary. g. Vertical angles are complementary. h. The two angles in a linear pair are both acute. i. If two angles form a linear pair, one is acute and the other is obtuse. 3. omplete each sentence. a. If two angles are supplementary and x is the measure of one of the angles, then the measure of the other angle is. b. If two angles are complementary and x is the measure of one of the angles, then the measure of the other angle is. Helping You emember 4. Look up the nonmathematical meaning of supplementary in your dictionary. How can this definition help you to remember the meaning of supplementary angles? Lesson -5 Glencoe/McGraw-Hill 9 Glencoe Geometry

33 -5 nrichment urve titching The star design at the right was created by a method known as curve stitching.lthough the design appears to contain curves, it is made up entirely of line segments. To begin the star design, draw a 60 angle. Mark eight equally-spaced points on each ray, and number the points as shown below. Then connect pairs of points that have the same number To make a complete star, make the same design in six 60 angles that have a common central vertex.. omplete the section of the star design above by connecting pairs of points that have the same number.. omplete the following design reate your own design. You may use several angles, and the angles may overlap. Glencoe/McGraw-Hill 30 Glencoe Geometry

34 -6 tudy Guide and Intervention olygons olygons polygon is a closed figure formed by a finite number of coplanar line segments. The sides that have a common endpoint must be noncollinear and each side intersects exactly two other sides at their endpoints. polygon is named according to its number of sides. regular polygon has congruent sides and congruent angles. polygon can be concave or convex. xample ame each polygon by its number of sides. Then classify it as concave or convex and regular or irregular. a. b. I H L Lesson -6 c. G The polygon has 4 sides, so it is a quadrilateral. It is concave because part of or lies in the interior of the figure. ecause it is concave, it cannot have all its angles congruent and so it is irregular. d. J K The figure is not closed, so it is not a polygon. The polygon has 5 sides, so it is a pentagon. It is convex. ll sides are congruent and all angles are congruent, so it is a regular pentagon. The figure has 8 congruent sides and 8 congruent angles. It is convex and is a regular octagon. xercises ame each polygon by its number of sides. Then classify it as concave or convex and regular or irregular Glencoe/McGraw-Hill 3 Glencoe Geometry

35 -6 tudy Guide and Intervention (continued) olygons erimeter The perimeter of a polygon is the sum of the lengths of all the sides of the polygon. There are special formulas for the perimeter of a square or a rectangle. xample Write an expression or formula for the perimeter of each polygon. ind the perimeter. a. 3 in. a 4 in. b c 5 in. a b c in. b. 5 cm s 5 cm s s 5 cm s 5 cm 4s 4(5) 0 cm c. ft w 3 ft w (3) () 0 ft w xercises ind the perimeter of each figure....5 cm 3 cm 5.5 ft 3.5 cm square yd 7 yd yd 4 yd 4 yd cm ind the length of each side of the polygon for the given perimeter x x x x x rectangle x Glencoe/McGraw-Hill 3 Glencoe Geometry

36 -6 kills ractice olygons ame each polygon by its number of sides and then classify it as convex or concave and regular or irregular Lesson ind the perimeter of each figure yd yd 0 yd 40 yd 4 m 6 m m 3 m 5 m 0 in. in. in. in. in. in. 0 in. in. OOIT GOMTY ind the perimeter of each polygon. 0. triangle with vertices (3, 5), (3, ), and (0, ). quadrilateral T with vertices ( 3, ), (, ), (, 4), and T( 3, 4). quadrilateral LMO with vertices L(, 4), M(3, 4), (, ), and O(, ) LG ind the length of each side of the polygon for the given perimeter millimeters kilometers feet 4w w Glencoe/McGraw-Hill 33 Glencoe Geometry

37 -6 ractice olygons ame each polygon by its number of sides and then classify it as convex or concave and regular or irregular ind the perimeter of each figure mm mm mi 33 mi 0 mm 8 mm 3 mi cm 6 cm 4 cm 4 cm 6 cm 6 cm 4 cm 4 cm OOIT GOMTY ind the perimeter of each polygon. 7. quadrilateral O with vertices O( 3, ), (, 5), (6, 4), and (5, ) 8. pentagon TUVW with vertices (0, 0), T(3, ), U(, 5), V(, 5), and W( 3, ) LG ind the length of each side of the polygon for the given perimeter inches centimeters. 89 feet 6n 8 n 3x 5 x 3 x 9 x 5x 4 WIG or xercises 3, use the following information. Jasmine plans to sew fringe around the scarf shown in the diagram.. How many inches of fringe does she need to purchase? 6 in. 4 in. 4 in. 6 in. 3. If Jasmine doubles the width of the scarf, how many inches of fringe will she need? Glencoe/McGraw-Hill 34 Glencoe Geometry

38 -6 eading to Learn Mathematics olygons re-ctivity How are polygons related to toys? ead the introduction to Lesson -6 at the top of page 45 in your textbook. ame four different shapes that can each be formed by four sticks connected to form a closed figure. ssume you have sticks with a good variety of lengths. Lesson -6 eading the Lesson. Tell why each figure is not a polygon. a. b. c.. ame each polygon by its number of sides. Then classify it as convex or concave and regular or not regular. a. b. c. 3. What is another name for a regular quadrilateral? 4. Match each polygon in the first column with the formula in the second column that can be used to find its perimeter. (s represents the length of each side of a regular polygon.) a. regular dodecagon i. 8s b. square ii. 6s c. regular hexagon iii. a b c d. rectangle iv. s e. regular octagon v. w f. triangle vi. 4s Helping You emember 5. One way to remember the meaning of a term is to explain it to another person. How would you explain to a friend what a regular polygon is? Glencoe/McGraw-Hill 35 Glencoe Geometry

39 -6 nrichment erimeter and rea of Irregular hapes Two formulas that are used frequently in mathematics are perimeter and area of a rectangle. erimeter: w rea: w, where is the length and w is the width However, many figures are combinations of two or more rectangles creating irregular shapes. To find the area of an irregular shape, it helps to separate the shape into rectangles, calculate the formula for each rectangle, then find the sum of the areas. xample ind the area of the figure at the right. eparate the figure into two rectangles. m 9 m 5 m w m 3 m The area of the irregular shape is 7 m. m 5 m 3 m ind the area and perimeter of each irregular shape in. 4 in. in. 6 cm cm 4 cm 4 cm cm 4 cm 4 cm 8 cm in m 3 m 7 ft 3 ft ft 9 ft m 9 m 6 m 6 m 6 ft 4 ft or xercises 5 8, find the perimeter of the figures in xercises escribe the steps you used to find the perimeter in xercise. Glencoe/McGraw-Hill 36 Glencoe Geometry

40 nswers (Lesson -) Lesson - - tudy Guide and Intervention oints, Lines, and lanes ame oints, Lines, and lanes In geometry, a point is a location, a line contains points, and a plane is a flat surface that contains points and lines. If points are on the same line, they are collinear. If points on are the same plane, they are coplanar. xample a. a line containing point Use the figure to name each of the following. The line can be named as. lso, any two of the three points on the line can be used to name it.,, or b. a plane containing point The plane can be named as plane or can be named using three noncollinear points in the plane, such as plane, plane, and so on. xercises efer to the figure.. ame a line that contains point.,,, or. What is another name for line m? m 3. ame a point not on. or 4. ame the intersection of and. 5. ame a point not on line or line m. raw and label a plane for each relationship. 6. is in plane. 7. T intersects at. T X Y 8. oint X is collinear with points and. nswers for xercises oint Y is not collinear with points T and. 0. Line contains points X and Y. Glencoe/McGraw-Hill Glencoe Geometry - tudy Guide and Intervention (continued) oints, Lines, and lanes oints, Lines, and lanes in pace pace is a boundless, three-dimensional set of all points. It contains lines and planes. xample a. How many planes appear in the figure? There are three planes: plane, plane O, and plane. O b. re points,, and coplanar? Yes. They are contained in plane O. xercises efer to the figure.. ame a line that is not contained in plane.. ame a plane that contains point. plane,plane, plane, plane, plane 3. ame three collinear points.,, efer to the figure. 4. How many planes are shown in the figure? 6 5. re points,, G, and H coplanar? xplain. o;, G, and H lie in plane GH, but does not. G J I H 6. ame a point coplanar with,, and. or J raw and label a figure for each relationship. 7. lanes M and intersect in HJ. 8. Line r is in plane, line s is in plane M, and lines r and s intersect at point J. t H J M s r 9. Line t contains point H and line t does not lie in plane M or plane. nswers for xercises 7 9 Glencoe/McGraw-Hill Glencoe Geometry Glencoe/McGraw-Hill Glencoe Geometry

41 - kills ractice oints, Lines, and lanes Glencoe/McGraw-Hill 3 Glencoe Geometry nswers Lesson - efer to the figure.. ame a line that contains point. p or p G n. ame a point contained in line n. or 3. What is another name for line p? or 4. ame the plane containing lines n and p. ample answer: plane G raw and label a figure for each relationship. ample answers are given. 5. oint K lies on T. 6. lane J contains line s. K T s J 7. Y lies in plane and contains 8. Lines q and f intersect at point Z point, but does not contain point H. in plane U. Y H U q Z f efer to the figure. 9. How many planes are shown in the figure? 5 0. How many of the planes contain points and? W. ame four points that are coplanar.,,, or,,, or,,,. re points,, and coplanar? xplain. Yes; points,, and lie in plane W. efer to the figure.. ame a line that contains points T and. g, T, T,. ame a line that intersects the plane containing points,, and. j or MT 3. ame the plane that contains T and. ample answer: plane raw and label a figure for each relationship. ample answers are given. 4. K and G intersect at point M 5. line contains L( 4, 4) and M(, 3). Line in plane T. q is in the same coordinate plane but does not intersect LM. Line q contains point. M G y T K M q O x L efer to the figure. T 6. How many planes are shown in the figure? 6 7. ame three collinear points., X, M W X 8. re points,,, and W coplanar? xplain. o; sample answer: points,, and lie in plane, but point W does not. M VIULIZTIO ame the geometric term(s) modeled by each object tip of pin. TO strings plane and line point lines. a car antenna 3. a library card line and point plane Glencoe/McGraw-Hill 4 Glencoe Geometry j nswers (Lesson -) - ractice (verage) oints, Lines, and lanes M T g h Glencoe/McGraw-Hill 3 Glencoe Geometry

42 nswers (Lesson -) Lesson - - eading to Learn Mathematics oints, Lines, and lanes re-ctivity Why do chairs sometimes wobble? ead the introduction to Lesson - at the top of page 6 in your textbook. ind three pencils of different lengths and hold them upright on your desk so that the three pencil points do not lie along a single line. an you place a flat sheet of paper or cardboard so that it touches all three pencil points? yes How many ways can you do this if you keep the pencil points in the same position? one How will your answer change if there are four pencil points? ample answer: It may not be possible to place the paper to touch all four points. eading the Lesson. omplete each sentence. a. oints that lie on the same lie are called collinear points. b. oints that do not lie in the same plane are called noncoplanar points. c. There is exactly one line through any two points. d. There is exactly one plane through any three noncollinear points.. efer to the figure at the right. Indicate whether each statement is true or false. a. oints,, and are collinear. false b. The intersection of plane and line m is point. true c. Line and line m do not intersect. false U m d. oints,,and can be used to name plane U. false e. Line lies in plane. true 3. omplete the figure at the right to show the following relationship: Lines, m, and n are coplanar and lie in plane. Lines and m intersect at point. Line n intersects line m at, but does not intersect line. m n Helping You emember 4. ecall or look in a dictionary to find the meaning of the prefix co-.what does this prefix mean? How can it help you remember the meaning of collinear? ample answer: The prefix co- means together. The word collinear contains the word line, so collinear means together on a line. Glencoe/McGraw-Hill 5 Glencoe Geometry - nrichment oints and Lines on a Matrix matrix is a rectangular array of rows and columns. oints and lines on a matrix are not defined in the same way as in uclidean geometry. point on a matrix is a dot, which can be small or large. line on a matrix is a path of dots that line up. etween two points on a line there may or may not be other points. Three examples of lines are shown at the upper right. The broad line can be thought of as a single line or as two narrow lines side by side. ot-matrix printers for computers used dots to form characters. The dots are often called pixels. The matrix at the right shows how a dot-matrix printer might print the letter. nswers may vary. ample answers are shown. raw points on each matrix to create the given figures.. raw two intersecting lines that have. raw two lines that cross but have four points in common. no common points. 3. Make the number 0 (zero) so that it 4. Make the capital letter O so that it extends to the top and bottom sides extends to each side of the matrix. of the matrix. 5. Using separate grid paper, make dot designs for several other letters. Which were the easiest and which were the most difficult? ee students work. Glencoe/McGraw-Hill 6 Glencoe Geometry Glencoe/McGraw-Hill 4 Glencoe Geometry

43 - tudy Guide and Intervention Linear Measure and recision Glencoe/McGraw-Hill 7 Glencoe Geometry nswers Lesson - Measure Line egments part of a line between two endpoints is called a line segment. The lengths of M and are written as M and. When you measure a segment, the precision of the measurement is half of the smallest unit on the ruler. xample xample ind the length of M. M ind the length of. cm 3 4 in. The long marks are centimeters, and the shorter marks are millimeters. The length of M is 3.4 centimeters. The measurement is accurate to within 0.5 millimeter, so M is between 3.35 centimeters and 3.45 centimeters long. The long marks are inches and the short marks are quarter inches. The length of is about 3 inches. The measurement is 4 accurate to within one half of a quarter inch, or 8 inch, so is between 5 inches and 8 7 inches long. 8 xercises ind the length of each line segment or object...5 cm. T in. 4 cm 3 in. 3. in cm 4 in. cm 3 ind the precision for each measurement in mm cm in. 0.5 mm 0.5 cm 8. ft mm 0. yd ft or 6 in mm yd or 9 in. 4 - tudy Guide and Intervention (continued) Linear Measure and recision alculate Measures On, to say that point M is between points and means,, and M are collinear and M M. On, 3 cm. We can say that the segments are congruent, or. lashes on the figure indicate which segments are congruent. M xample ind. xample ind x and.. cm.9 cm alculate by adding and Therefore, is 3. centimeters long. x 5 x x is between and. x x x 5 3x x 5 x 5 x 5 (5) 5 5 xercises ind the measurement of each segment. ssume that the art is not drawn to scale.. T 4.5 cm. 3.0 cm.5 cm 6 in. in. 4 T 3 4 in. 3. X Z in. in. 4 X Y Z in. 4. W X 3 cm 6 cm W X Y ind x and if is between and T. 5. 5x, T 3x, and T 48. 6, x, T 5x 4, and T 3. 4, x, T, and T 7. 0, x, T, and T 4. 3, Use the figures to determine whether each pair of segments is congruent. 9. and yes 0. X Y and Y Z no cm 5 cm 5 cm cm X 3x 5 5x Y Z 9x Glencoe/McGraw-Hill 8 Glencoe Geometry nswers (Lesson -) Glencoe/McGraw-Hill 5 Glencoe Geometry

44 nswers (Lesson -) Lesson - - kills ractice Linear Measure and recision ind the length of each line segment or object... cm in. about 55 mm about in. 4 ind the precision for each measurement feet 4. centimeters 5. 9 inches 0.5 ft 0.5 cm in. 4 ind the measurement of each segment G H in. 4 in. 4.9 cm 5. cm 9.7 mm G H 5 mm in. 0. cm 5.3 mm 4 ind the value of the variable and YZ if Y is between X and Z. 9. XY 5p, YZ p, and XY 5 0. XY, YZ g, and XZ 8 5; 5 8; 6. XY 4m, YZ 3m, and XZ 4. XY c, YZ 6c, and XZ 8 6; 8 0; 60 Use the figures to determine whether each pair of segments is congruent. 3., 4. M, 5. W X, W Z m 3 m 3 m 5 m yd M 0 yd yes no no Y 9 ft Z 5 ft 5 ft yd X W Glencoe/McGraw-Hill 9 Glencoe Geometry - ractice (verage) Linear Measure and recision ind the length of each line segment or object... in. cm in. 4 mm 6 ind the precision for each measurement meters 4. 7 inches millimeters m in. 0.5 mm 8 ind the measurement of each segment W X 8.4 cm 4.7 cm 3. cm 3 5 in. 0.4cm in. 4 in. W X 89.6 cm Y 00 cm ind the value of the variable and KL if K is between J and L. 9. JK 6r, KL 3r, and JL 7 0. JK s, KL s, and JL 5s 0 3; 9 6; 8 Use the figures to determine whether each pair of segments is congruent.. T U, W., 3. G, T ft.7 in. ft 3 ft U W 3 ft.9 in. G 5x H 6x no yes no 4. TY Jorge used the figure at the right to make a pattern for a mosaic he plans to inlay on a tabletop. ame all of the congruent segments in the figure., Glencoe/McGraw-Hill 0 Glencoe Geometry Glencoe/McGraw-Hill 6 Glencoe Geometry

45 nswers (Lesson -) - eading to Learn Mathematics Linear Measure and recision Glencoe/McGraw-Hill Glencoe Geometry nswers Lesson - re-ctivity Why are units of measure important? ead the introduction to Lesson - at the top of page 3 in your textbook. The basic unit of length in the metric system is the meter. How many meters are there in one kilometer? 000 o you think it would be easier to learn the relationships between the different units of length in the customary system (used in the United tates) or in the metric system? xplain your answer. ample answer: The metric system is easier because you can change between the different units by just moving the decimal point. eading the Lesson. xplain the difference between a line and a line segment and why one of these can be measured, while the other cannot. ample answer: line is infinite. ince it has no endpoints, a line does not have a definite length and cannot be measured. line segment has two endpoints, so it has a definite length and can be measured.. What is the smallest length marked on a -inch ruler? ample answer: in. 6 What is the smallest length marked on a centimeter ruler? mm 3. ind the precision of each measurement. a. 5 cm 0.5 cm b. 5.0 cm 0.05 cm 4. efer to the figure at the right. Which one of the following statements is true? xplain your answer. ; ample answer: The two segments are congruent because they have the same measure or length. They are not equal because they are not the same segment. 4.5 cm 4.5 cm 5. uppose that is a point on V W and is not the same point as V or W.Tell whether each of the following statements is always, sometimes, or never true. a. V W sometimes b. is between V and W. always c. V VW W never Helping You emember 6. good way to remember terms used in mathematics is to relate them to everyday words you know. Give three words that are used outside of mathematics that can help you remember that there are 00 centimeters in a meter. ample answer: cent, century, centennial - nrichment oints quidistant from egments The distance from a point to a segment is zero if the point is on the segment. Otherwise, it is the length of the shortest segment from the point to the segment. figure is a locus if it is the set of all points that satisfy a set of conditions. The locus of all points that are inch 4 from the segment is shown by two dashed segments with semicircles at both ends.. uppose,,, and are four different points, and consider the locus of all points x units from and x units from. Use any unit you find convenient. The locus can take different forms. ketch at least three possibilities. List some of the things that seem to affect the form of the locus. ample answers are shown. X Y X Y The locus is The locus is a set of The locus is a pair segment X Y midway points, X and Y. of line segments, between and. and. The locus of points x units from and depends on the distance x and how and are situated relative to one another.. onduct your own investigation of the locus of points equidistant from two segments. escribe your results on a separate sheet of paper. ee students work. Glencoe/McGraw-Hill Glencoe Geometry Glencoe/McGraw-Hill 7 Glencoe Geometry

46 nswers (Lesson -3) Lesson -3-3 tudy Guide and Intervention istance and Midpoints istance etween Two oints istance on a umber Line istance in the oordinate lane a b b a or a b ythagorean Theorem: a b c istance ormula: d (x x ) (y y ) (, ) O y (, 3) x (, ) xample ind. xample ind the distance between (, ) and (, 3) ( 4) 6 6 ythagorean Theorem () () () () (3) (4) () istance ormula d (x x ) (y y ) ( )) ( (3 ( )) (3) (4) 5 5 xercises Use the number line to find each measure. G. 6. G G 5 6. G Use the ythagorean Theorem to find the distance between each pair of points. 9. (0, 0), (6, 8) 0 0. (, 3), (3, 5) 3. M(, ), (9, 3) 7. (, ), ( 9, 6) 5 Use the istance ormula to find the distance between each pair of points. 3. (0, 0), (5, 0) 5 4. O(, 0), ( 8, 3) 5 5. (, ), (6, ) (, 0), ( 4, 3) Glencoe/McGraw-Hill 3 Glencoe Geometry -3 tudy Guide and Intervention (continued) istance and Midpoints Midpoint of a egment Midpoint on a If the coordinates of the endpoints of a segment are a and b, umber Line then the coordinate of the midpoint of the segment is a. b If a segment has endpoints with coordinates (x Midpoint on a, y ) and (x, y ), oordinate lane then the coordinates of the midpoint of the segment are x x, y y. xample ind the coordinate of the midpoint of. 3 0 The coordinates of and are 3 and. If M is the midpoint of, then the coordinate of M is 3 or. xample M is the midpoint of for (, 4) and (4, ). ind the coordinates of M. M x x, y y 4, 4 or (,.5) xercises Use the number line to find the coordinate of the midpoint of each segment... G 4 G G G ind the coordinates of the midpoint of a segment having the given endpoints. 9. (0, 0), (, 8) (6, 4) 0. (, 8), (6, ) ( 3, 0). M(, ), ( 9, 3) (, 5.5). (, 6), ( 9, 3) ( 5.5, 4.5) 3. (0, ), T(9, 0) (9.5, 6) 4. M(, ), ( 9, 6) ( 5, 4) Glencoe/McGraw-Hill 4 Glencoe Geometry Glencoe/McGraw-Hill 8 Glencoe Geometry