2. Welcome to Interactive Chalkboard Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

3. Splash Screen

4. Contents Lesson 1-1Points, Lines, and Planes Lesson 1-2Linear Measure and Precision Lesson 1-3Distance and Midpoints Lesson 1-4Angle Measure Lesson 1-5Angle Relationships Lesson 1-6Polygons

5. Lesson 1 Contents Example 1Name Lines and Planes Example 2Model Points, Lines, and Planes Example 3Draw Geometric Figures Example 4Interpret Drawings

6. Example 1-1a Use the figure to name a line containing point K. Answer: The line can be named as line a. There are three points on the line. Any two of the points can be used to name the line.

7. Example 1-1b Use the figure to name a plane containing point L. Answer: The plane can be named as plane B. You can also use the letters of any three noncollinear points to name the plane. plane JKM plane KLM plane JLM

8. Example 1-1c The letters of each of these names can be reordered to create other acceptable names for this plane. For example, JKM can also be written as JMK, MKJ, KJM, KMJ, and MJK. There are 18 different three-letter names for this plane.

9. Example 1-1d Answer: line c, Answer:plane P, plane XYZ, plane ZYX, plane YZX, plane XZY, plane ZXY, plane YXZ Use the figure to name each of the following. a. a line containing point X b. a plane containing point Z

10. Example 1-2a VISUALIZATION Name the geometric shape modeled by the long hand on a clock. Answer: The long hand on a clock models a line segment.

11. Example 1-2b VISUALIZATION Name the geometric shape modeled by a 10  12 patio. Answer: The patio models a plane.

12. Example 1-2c VISUALIZATION Name the geometric shape modeled by the location where the corner of a driveway meets the road. Answer: The location where the corner of a driveway meets the road models a point.

13. Example 1-2d Answer: point Answer: plane VISUALIZATION Name the geometric shape modeled by each object. a. a colored dot on a map used to mark the location of a city b. the ceiling of your classroom c. the railing on a stairway Answer: line segment

14. Example 1-3a Draw a surface to represent plane R and label it. Draw and label a figure for the following situation. Plane R contains lines and, which intersect at point P. Add point C on plane R so that it is not collinear with or.

15. Example 1-3b Draw a line anywhere on the plane. Draw and label a figure for the following situation. Plane R contains lines and, which intersect at point P. Add point C on plane R so that it is not collinear with or.

16. Example 1-3c Draw dots on the line for points A and B. Label the points. B A Draw and label a figure for the following situation. Plane R contains lines and, which intersect at point P. Add point C on plane R so that it is not collinear with or.

17. Example 1-3d B A Draw a line intersecting. Draw and label a figure for the following situation. Plane R contains lines and, which intersect at point P. Add point C on plane R so that it is not collinear with or.

18. E D Example 1-3e Draw dots on this line for points D and E. Label the points. B A Draw and label a figure for the following situation. Plane R contains lines and, which intersect at point P. Add point C on plane R so that it is not collinear with or.

19. B A E D Example 1-3f P Label the intersection point of the two lines as P. Draw and label a figure for the following situation. Plane R contains lines and, which intersect at point P. Add point C on plane R so that it is not collinear with or.

20. B A E D Example 1-3g P C Draw a dot for point C in plane R such that it will not lie on or. Label the point. Answer: Draw and label a figure for the following situation. Plane R contains lines and, which intersect at point P. Add point C on plane R so that it is not collinear with or.

21. Example 1-3h Graph each point and draw. There are an infinite number of points that are collinear with Q and R. In the graph, one such point is T(1, 0). on a coordinate plane contains Q(–2, 4) and R(4, –4). Add point T so that T is collinear with these points. Draw and label a figure for the following situation.

22. Example 1-3i Draw and label a figure for each relationship. a. Plane D contains line a, line m, and line t, with all three lines intersecting at point Z. Add point F on plane D so that it is not collinear with any of the three given lines. Sample answer:

23. Example 1-3j Sample answer: b. on a coordinate plane contains B(–3, –2) and A(3, 2). Add point M so that M is collinear with these points. Draw and label a figure for each relationship.

24. Example 1-4a How many planes appear in this figure? Answer: There are two planes: plane S and plane ABC.

25. Example 1-4b Name three points that are collinear. Answer: Points A, B, and D are collinear.

26. Example 1-4c Are points A, B, C, and D coplanar? Explain. Answer: Points A, B, C, and D all lie in plane ABC, so they are coplanar.

27. Example 1-4d Answer: The two lines intersect at point A. At what point do and intersect?

28. Example 1-4e a. How many planes appear in this figure? Answer: two

29. Example 1-4f b. Name three points that are collinear. Sample answer: A, X, and Z

30. Example 1-4g c. Are points X, O, and R coplanar? Explain. Answer: Points X, O, and R all lie in plane T, so they are coplanar.

31. Example 1-4h Answer: R d. At what point do and intersect?

32. End of Lesson 1

33. Lesson 2 Contents Example 1Length in Metric Units Example 2Length in Customary Units Example 3Precision Example 4Find Measurements Example 5Congruent Segments

34. Example 2-1a Find the length of. The long marks are centimeters, and the shorter marks are millimeters. There are 10 millimeters for each centimeter. Answer: is about 42 millimeters long.

35. Find the length of. Example 2-1b The ruler is marked in centimeters. Point R is closer to the 5-centimeter mark than to 4 centimeters. Answer: is about 5 centimeters long.

36. Example 2-1c Answer: 18 mm a. Find the length of

37. Example 2-1d Answer: 2 cm b. Find the length of

38. Example 2-2a Find the length of. Each inch is divided into sixteenths. Point E is closer to the 3-inch mark. Answer: is about 3 inches long.

39. Find the length of. Example 2-2b Each inch is divided into fourths. Point G is closer to the -inch mark. Answer: is about inches long.

40. Example 2-2c a. Find the length of. Answer: in.

41. b. Find the length of. Example 2-2d Answer: in.

42. Example 2-3a Find the precision for inches. Explain its meaning. Answer: The precision is inch. The measurement could be inches to inches. The measuring tool is divided into -inch increments. Thus, the measurement is precise to within inch.

43. Example 2-3b Find the precision for 15 millimeters. Explain its meaning. Answer:The precision is 0.5 millimeter. The measurement could be 14.5 millimeters to 15.5 millimeters. The measuring tool is divided into millimeter increments. Thus, the measurement is precise to within (1) or 0.5 millimeter.

44. Example 2-3c Answer: The precision is 0.5 millimeter. The measurement could be 87.5 millimeters to 88.5 millimeters. Answer: The precision is inch. The measurement could be inches to inches. PRECISION Find the precision for each measurement. Explain its meaning. a. 88 millimeters b.

45. Example 2-4a Find LM. Point M is between L and N. Sum of parts whole Substitution Subtract 2.6 from each side. Simplify. Answer: is 1.4 centimeters long. LM is the measure of.

46. XZ is the measure of. Example 2-4b Find XZ. Point Y is between X and Z. XZ can be found by adding XY and YZ. Sum of parts whole Substitution Add.

47. Example 2-4c Answer: is inches long.

48. Example 2-4d Substitute known values. Add 3 to each side. Divide each side by 12. Simplify. Find x and ST if T is between S and U, ST 7x, SU 45, and TU 5x – 3.

49. Example 2-4e Given Multiply. Answer:

50. Example 2-4f a. Find SE. Answer:in.

51. Example 2-4g b. Find ON. Answer: 3.7 cm

52. Example 2-4h c. Find a and AB if AB 4a + 10, BC 3a – 5, and AC 19. Answer:

53. Example 2-5a FONTS The Arial font is often used because it is easy to read. Study the word time shown in Arial type. Each letter can be broken into individual segments. The letter T would have two segments, a short horizontal segment, and a long vertical segment. Assume that all segments overlap when they meet. Which segments are congruent? TIME Answer: The five vertical segments in the letters T, I, M, and E are congruent. The four horizontal segments in T and E are congruent. The two diagonal segments in the letter M are congruent.

54. Example 2-5b LEISURE ACTIVITIES The graph shows the percent of adults who participated in selected activities. Suppose a segment was drawn along the height of each bar. Which categories would have segments that are congruent?

55. Example 2-5c Answer: The segments on the bars for going to museums and picnics would be congruent because they both have the same height, representing 16%. The segments on bars for going to the zoo and playing board games would be congruent because they have the same height, representing 14%.

56. End of Lesson 2

57. Lesson 3 Contents Example 1Find Distance on a Number Line Example 2Find Distance on a Coordinate Plane Example 3Find Coordinates of Midpoint Example 4Find Coordinates of Endpoint Example 5Use Algebra to Find Measures

58. Example 3-1a Use the number line to find QR. The coordinates of Q and R are –6 and –3. Answer: 3 Distance Formula Simplify.

59. Example 3-1b Use the number line to find AX. Answer: 8

60. Example 3-2a Find the distance between E(–4, 1) and F(3, –1). Method 1 Pythagorean Theorem Use the gridlines to form a triangle so you can use the Pythagorean Theorem.

61. Example 3-2b Pythagorean Theorem Simplify. Take the square root of each side.

62. Example 3-2c Method 2 Distance Formula Distance Formula Simplify. Answer: The distance from E to F is units. You can use a calculator to find that is approximately 7.28.

63. Example 3-2d Find the distance between A(–3, 4) and M(1, 2). Answer:

64. Example 3-3a The coordinates of J and K are –12 and 16. Answer: 2 Simplify. The coordinates on a number line of J and K are –12 and 16, respectively. Find the coordinate of the midpoint of. Let M be the midpoint of.

65. Let G be and H be. Example 3-3b Answer: (–3, 3) Find the coordinates of M, the midpoint of, for G(8, –6) and H(–14, 12).

66. a.The coordinates on a number line of Y and O are 7 and –15, respectively. Find the coordinate of the midpoint of. b.Find the coordinates of the midpoint of for X(–2, 3) and Y(–8, –9). Example 3-3c Answer: (–5, –3) Answer: –4

67. Let F be in the Midpoint Formula. Example 3-4a Find the coordinates of D if E(–6, 4) is the midpoint of and F has coordinates (–5, –3). Write two equations to find the coordinates of D.

68. Example 3-4b Solve each equation. Answer: The coordinates of D are (–7, 11). Multiply each side by 2. Add 5 to each side. Multiply each side by 2. Add 3 to each side.

69. Example 3-4c Answer: (17, –11) Find the coordinates of R if N(8, –3) is the midpoint of and S has coordinates (–1, 5).

70. Example 3-5a Multiple-Choice Test Item What is the measure of if Q is the midpoint of ? A B 4C D 9

71. Example 3-5b Read the Test Item Solve the Test Item Because Q is the midpoint, you know that. Use this equation and the algebraic measures to find a value for x. You know that Q is the midpoint of, and the figure gives algebraic measures for and. You are asked to find the measure of.

72. Example 3-5c Definition of midpoint Distributive Property Subtract 1 from each side. Add 3x to each side. Divide each side by 10.

73. Example 3-5d Answer: D Original measure Simplify. Now substitute for x in the expression for PR.

74. Example 3-5e Answer: B Multiple-Choice Test Item What is the measure of if B is the midpoint of ? A 1B 3C 5D 10

75. End of Lesson 3

76. Lesson 4 Contents Example 1Angles and Their Parts Example 2Measure and Classify Angles Example 3Use Algebra to Find Angle Measures

77. Example 4-1a Name all angles that have B as a vertex. Answer:  5,  6,  7, and  ABG

78. Example 4-1b Name the sides of  5. Answer: and or are the sides of  5.

79. Example 4-1c Write another name for  6. Answer:  EBD,  FBD,  DBF, and  DBE are other names for  6.

80. Example 4-1d a. Name all angles that have X as a vertex. b. Name the sides of  3. c. Write another name for  3. Answer:  1,  2,  3, and  RXB or  RXN Answer:  AXB,  AXN,  NXA,  BXA Answer:

81. Example 4-2a Measure  TYV and classify it as right, acute, or obtuse.  TYV is marked with a right angle symbol, so measuring is not necessary. Answer: is a right angle.

82. Example 4-2b Measure  WYT and classify it as right, acute, or obtuse. Use a protractor to find that. Answer: > is an obtuse angle.

83. Example 4-2c Measure  TYU and classify it as right, acute, or obtuse. Use a protractor to find that m. Answer: is an acute angle.

84. Example 4-2d Measure each angle named and classify it as right, acute, or obtuse. a.  CZD b.  CZE c.  DZX Answer: 150, obtuse Answer: 90, right Answer: 30, acute

85. Example 4-3a INTERIOR DESIGN Wall stickers of standard shapes are often used to provide a stimulating environment for a young child’s room. A five-pointed star sticker is shown with vertices labeled. Find m  GBH and m  HCI if  GBH  HCI, m  GBH 2x + 5, and m  HCI 3x – 10.

86. Example 4-3b Given Definition of congruent angles Substitution Add 10 to each side. Subtract 2x from each side.

87. Example 4-3c Given Simplify. Use the value of x to find the measure of one angle. Since. or 35 Answer: Both measure 35.

88. Example 4-3d SIGNS A railroad crossing sign forms congruent angles. In the figure,  WVX  ZVY. If m  WVX 7a + 13 and m  ZVY 10a – 20, find the actual measurements of  WVX and  ZVY. Answer:

89. End of Lesson 4

90. Lesson 5 Contents Example 1Identify Angle Pairs Example 2Angle Measure Example 3Perpendicular Lines Example 4Interpret Figures

91. Example 5-1a Name two angles that form a linear pair. A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays. Answer: The angle pairs that satisfy this definition are

92. Example 5-1b Name two acute vertical angles. There are four acute angles shown. There is one pair of vertical angles. Answer: The acute vertical angles are  VZY and  XZW.

93. Example 5-1c Name an angle pair that satisfies each condition. a. two acute vertical angles b.two adjacent angles whose sum is less than 90 Answer:  BAC and  CAD or  EAF and  FAN Answer:  BAC and  FAE,  CAD and  NAF, or  BAD and  NAE

94. Example 5-2a ALGEBRA Find the measures of two supplementary angles if the measure of one angle is 6 less than five times the other angle. ExploreThe problem relates the measures of two supplementary angles. You know that the sum of the measures of supplementary angles is 180. PlanDraw two figures to represent the angles.

95. Example 5-2b Let the measure of one angle be x. Solve Given Simplify. Add 6 to each side. Divide each side by 6.

96. Example 5-2c Use the value of x to find each angle measure. ExamineAdd the angle measures to verify that the angles are supplementary. Answer: 31, 149

97. Example 5-2d ALGEBRA Find the measures of two complementary angles if one angle measures six degrees less than five times the measure of the other. Answer: 16, 74

98. Example 5-3a ALGEBRA Find x so that.

99. Example 5-3b If, then m  KJH 90. To find x, use  KJI and  IJH. Substitution Add. Subtract 6 from each side. Divide each side by 12. Answer: Sum of parts whole

100. Example 5-3c ALGEBRA Find x and y so that and are perpendicular. Answer:

101. Example 5-4a The diagram is marked to show that From the definition of perpendicular, perpendicular lines intersect to form congruent adjacent angles. Answer: Yes; and are perpendicular. Determine whether the following statement can be assumed from the figure below. Explain. m  VYT 90

102. Example 5-4b Determine whether the following statement can be assumed from the figure below. Explain.  TYW and  TYU are supplementary. Answer: Yes; they form a linear pair of angles.

103. Example 5-4c Determine whether the following statement can be assumed from the figure below. Explain.  VYW and  TYS are adjacent angles. Answer: No; they do not share a common side.

104. Determine whether each statement can be assumed from the figure below. Explain. a. b.  TAU and  UAY are complementary. c.  UAX and  UXA are adjacent. Example 5-4d Answer: Yes; lines TY and SX are perpendicular. Answer: No; they do not share a common side. Answer: No; the sum of the two angles is 180, not 90.

105. End of Lesson 5

106. Lesson 6 Contents Example 1Identify Polygons Example 2Find Perimeter Example 3Perimeter on the Coordinate Plane Example 4Use Perimeter to Find Sides

107. Example 6-1a Name the polygon by its number of sides. Then classify it as convex or concave, regular or irregular. There are 4 sides, so this is a quadrilateral. No line containing any of the sides will pass through the interior of the quadrilateral, so it is convex. The sides are not congruent, so it is irregular. Answer: quadrilateral, convex, irregular

108. Example 6-1b Name the polygon by its number of sides. Then classify it as convex or concave, regular or irregular. There are 9 sides, so this is a nonagon. A line containing some of the sides will pass through the interior of the nonagon, so it is concave. The sides are not congruent, so it is irregular. Answer: nonagon, concave, irregular

109. Example 6-1c Answer: triangle, convex, regular Answer: quadrilateral, convex, irregular Name each polygon by the number of sides. Then classify it as convex or concave, regular or irregular. a. b.

110. Example 6-2a CONSTRUCTION A masonry company is contracted to lay three layers of decorative brick along the foundation for a new house given the dimensions below. Find the perimeter of the foundation and determine how many bricks the company will need to complete the job. Assume that one brick is 8 inches long.

111. Example 6-2b First, find the perimeter. Add the lengths of the sides. The perimeter of the foundation is 216 feet.

112. Example 6-2c Next, determine how many bricks will be needed to complete the job. Each brick measures 8 inches, or foot. Divide 216 by to find the number of bricks needed for one layer. Answer: The builder will need 324 bricks for each layer. Three layers of bricks are needed, so the builder needs 324 3 or 972 bricks.

113. Example 6-2d CONSTRUCTION The builder realizes he accidentally halved the size of the foundation in part a. How will this affect the perimeter of the house and the number of bricks the masonry company needs?

114. Example 6-2e The new dimensions are twice the measures of the original lengths. The perimeter has doubled. Answer: The perimeter and the number of bricks needed are doubled. The new number of bricks needed for one layer is or 648. For three layers, the total number of bricks is 648 3 or 1944 bricks.

115. Example 6-2f SEWING Miranda is making a very unusual quilt. It is in the shape of a hexagon as shown below. She wants to trim the edge with a special blanket binding. The binding is sold by the yard. a.Find the perimeter of the quilt in inches. Then determine how many yards of binding Miranda will need for the quilt. Answer: 336 in., yd

116. Example 6-2f SEWING Miranda is making a very unusual quilt. It is in the shape of a hexagon as shown below. She wants to trim the edge with a special blanket binding. The binding is sold by the yard. b.Miranda decides to make four quilts. How will this affect the amount of binding she will need? How much binding will she need for this project? Answer:The amount of binding is multiplied by 4. She will need yards.

117. Example 6-3a Find the perimeter of pentagon ABCDE with A(0, 4), B(4, 0), C(3, –4), D(–3, –4), and E(–3, 1).

118. Example 6-3b Use the Distance Formula,, to find AB, BC, CD, DE, and EA.

119. Example 6-3c Answer: The perimeter of pentagon ABCDE is or about 25 units.

120. Example 6-3d Find the perimeter of quadrilateral WXYZ with W(2, 4), X(–3, 3), Y(–1, 0), and Z(3, –1). Answer: about 17.9 units

121. Example 6-4a The width of a rectangle is 5 less than twice its length. The perimeter is 80 centimeters. Find the length of each side. Let represent the length. Then the width is.

122. Example 6-4b Perimeter formula for rectangle Multiply. Simplify. Add 10 to each side. Divide each side by 6. Answer: The length is 15 cm. By substituting 15 for, the width becomes 2(15) – 5 or 25 cm.

123. Example 6-4c The length of a rectangle is 7 more than five times its width. The perimeter is 134 feet. Find the length of each side. Answer:

124. End of Lesson 6

125. Online Explore online information about the information introduced in this chapter. Click on the Connect button to launch your browser and go to the Glencoe Geometry Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to www.geometryonline.com/extra_examples.

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