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GUIDED PRACTICE Vocabulary Check ✓ Concept Check ✓ Skill Check ✓ 3 APPLY G B 1. Name an interior angle and an exterior angle of the polygon shown at the right. ASSIGNMENT GUIDE C A BASIC ™A, ™B, ™BCD, ™D, or ™AED; ™AEF, ™BCG, or ™DCH F E H D Day 1: pp. 665–666 Exs. 6–40 even Day 2: pp. 665–668 Exs. 7–41 odd, 49–56, 58–61, 63–73 odd 2. How many exterior angles are there in an n-gon? Are they all considered when using the Polygon Exterior Angles Theorem? Explain. 2n (2 at each vertex); no, only one at each vertex Find the value of x. 3. 120ⴗ 95 4. 120 AVERAGE 5. Day 1: pp. 665–666 Exs. 6–40 even Day 2: pp. 665–668 Exs. 7–43 odd, 49–61, 63–73 odd 45 105ⴗ 115ⴗ xⴗ 105ⴗ ADVANCED xⴗ xⴗ Day 1: pp. 665–666 Exs. 6–40 even Day 2: pp. 665–668 Exs. 7–41 odd, 43–45, 49–62, 63–73 odd PRACTICE AND APPLICATIONS STUDENT HELP Extra Practice to help you master skills is on p. 823. BLOCK SCHEDULE SUMS OF ANGLE MEASURES Find the sum of the measures of the interior angles of the convex polygon. 6. 10-gon 1440° 10. 20-gon 3240° 7. 12-gon 1800° 11. 30-gon 5040° 8. 15-gon 9. 18-gon 2880° 2340° 12. 40-gon 6840° 13. 100-gon 17,640° ANGLE MEASURES In Exercises 14–19, find the value of x. 14. 127 xⴗ 15. 101 113ⴗ 146ⴗ 106ⴗ 147ⴗ 80ⴗ 16. 80 102ⴗ 98ⴗ 120ⴗ 143ⴗ 124ⴗ 170ⴗ 125ⴗ xⴗ 130ⴗ 17. 108 xⴗ 18. 158ⴗ xⴗ xⴗ STUDENT HELP HOMEWORK HELP Example 1: Exs. 6–16, 20, 21 Example 2: Exs. 17–19, 22–28 Example 3: Exs. 29–38 Example 4: Exs. 39, 40, 49, 50 Example 5: Exs. 51–54 about 128.6 20. A convex quadrilateral has interior angles that measure 80°, 110°, and 80°. What is the measure of the fourth interior angle? 90° 21. A convex pentagon has interior angles that measure 60°, 80°, 120°, and 140°. What is the measure of the fifth interior angle? 140° DETERMINING NUMBER OF SIDES In Exercises 22–25, you are given the measure of each interior angle of a regular n-gon. Find the value of n. 22. 144° 10 23. 120° 6 24. 140° 9 25. 157.5° 16 11.1 Angle Measures in Polygons EXERCISE LEVELS Level A: Easier 6–13 Level B: More Difficult 14–42, 47–61 Level C: Most Difficult 43–46, 62 HOMEWORK CHECK To quickly check student understanding of key concepts, go over the following exercises: Exs. 8, 14, 20, 22, 26, 30, 34, 38, 51, 56. See also the Daily Homework Quiz: • Blackline Master (Chapter 11 Resource Book, p. 25) • Transparency (p. 80) 19. 135 xⴗ pp. 665–668 Exs. 6–41, 43, 49–61, 63–73 odd COMMON ERROR EXERCISES 22–25 Students may make some algebraic errors when solving these problems. Point out that after they set up the equation, they should multiply both sides by n. 665 665 42. According to the Polygon Interior Angles Thm., the sum of the measures of the interior √ of a convex polygon is dependent only on the number of sides. Therefore, the sums of the measures of the interior √ of any two polygons with the same number of sides are equal. It does not matter whether one or both of the polygons are regular or whether the two are similar. FOCUS ON APPLICATIONS L AL I FE 39. The yellow hexagon is regular with interior angles measuring 120° each; the yellow pentagons each have two interior angles that measure 90° and three interior angles that measure 120°; the triangles are equilateral with all interior angles measuring 60°. 40. The yellow decagon is regular with each interior angle measuring 144°; the yellow pentagons each have two interior angles measuring 90°, two interior angles measuring 108° and one interior angle measuring 144°; the red quadrilaterals have one angle measuring 144° and three angles measuring 72° each. 43. Draw all the diagonals of ABCDE that have A as one endpoint. The diagonals, A 苶C苶 and A 苶D 苶, divide ABCDE into 3 †s . By the Angle Addition Post., måBAE = måBAC + måCAD + måDAE. Similarly, måBCD = måBCA + måACD and måCDE = måCDA + måADE. Then the sum of the measures of the interior ås of ABCDE is equal to the sum of the measures of ås of †ABC, †ACD, and †ADE. By the † Sum Thm., the sum of the measures of each † is 180°, so the sum of the measures of the interior ås of ABCDE is 3 ⴢ 180° = (5 – 2) ⴢ 180°. RE COMMON ERROR EXERCISES 29–38 Students may use the formulas for interior angles to do these exercises. Point out that if they are using exterior angles, they should be using the formulas on page 663 not the ones on page 662. 41. ™3 and ™8 are a linear pair, so m™3 = 140°; ™2 and ™7 are a linear pair, so m™7 = 80°; m™1 = 80° by the Polygon Interior Angles Thm.; ™1 and ™6 are a linear pair, so m™6 = 100°; ™4 and ™9 are a linear pair, as are ™5 and ™10, so m™9 = m™10 = 70°. STAINED GLASS is tinted glass that has been cut into shapes and arranged to form a picture or design. The pieces of glass are held in place by strips of lead. 44. Let A be a regular n-gon and x° the measure of each interior ™. By the Polygon Interior Angles Thm., the sum of the measures of the interior √ of A is (n – 2) • 180°. That is, n • x° = (n – 2) • 180°, or x° = (n º 2) • 180° } }. n 666 666 CONSTRUCTION Use a compass, protractor, and ruler to check the results of Example 2 on page 662. 26–28. Check drawings. 26. Draw a large angle that measures 140°. Mark congruent lengths on the sides of the angle. 27. From the end of one of the congruent lengths in Exercise 26, draw the second side of another angle that measures 140°. Mark another congruent length along this new side. 28. Continue to draw angles that measure 140° until a polygon is formed. Verify that the polygon is regular and has 9 sides. DETERMINING ANGLE MEASURES In Exercises 29–32, you are given the number of sides of a regular polygon. Find the measure of each exterior angle. 29. 12 30° 30. 11 about 32.7° 31. 21 about 17.1° 32. 15 24° DETERMINING NUMBER OF SIDES In Exercises 33–36, you are given the measure of each exterior angle of a regular n-gon. Find the value of n. 33. 60° 6 34. 20° 18 35. 72° 36. 10° 5 36 37. A convex hexagon has exterior angles that measure 48°, 52°, 55°, 62°, and 68°. What is the measure of the exterior angle of the sixth vertex? 75° 38. What is the measure of each exterior angle of a regular decagon? 36° STAINED GLASS WINDOWS In Exercises 39 and 40, the purple and green pieces of glass are in the shape of regular polygons. Find the measure of each interior angle of the red and yellow pieces of glass. 39, 40. See margin. 39. 40. 41. FINDING MEASURES OF ANGLES In the diagram at the right, m™2 = 100°, m™8 = 40°, m™4 = m™5 = 110°. Find the measures of the other labeled angles and explain your reasoning. See margin. 42. 8 9 7 2 3 4 5 1 6 10 Writing Explain why the sum of the measures of the interior angles of any two n-gons with the same number of sides (two octagons, for example) is the same. Do the n-gons need to be regular? Do they need to be similar? See margin. 43. PROOF Use ABCDE to write a paragraph proof to prove Theorem 11.1 for pentagons. See margin. 44. PROOF Use a paragraph proof to prove the Corollary to Theorem 11.1. Chapter 11 Area of Polygons and Circles A E B C D 45. Let A be a convex n-gon. Each interior ™ and one of the exterior √ at that vertex form a linear pair, so the sum of their measures is 180°. Then the sum of the measures of the interior ™ and one exterior ™ at each vertex is n • 180°. By the Polygon Interior Angles Thm., the sum of the measures of the interior √ of A is (n – 2) • 180°. So the sum of the measures of the exterior √ of A, one at each vertex, is n • 180° – (n – 2) • 180° = n • 180° – n • 180° + 360° = 360°. 45. PROOF Use this plan to write a paragraph proof of Theorem 11.2. Plan for Proof In a convex n-gon, the sum of the measures of an interior (n – 2) ⴢ 180° n 1 n = 3 }}. It is not possible for a 3 1 polygon to have 3 }} sides. 3 (n – 2) ⴢ 180° 54. No; if }} = 18°, then n 2 n = 2 }}. It is not possible for a 9 2 polygon to have 2 }} sides. 9 53. No; if }} = 72°, then angle and an adjacent exterior angle at any vertex is 180°. Multiply by n to get the sum of all such sums at each vertex. Then subtract the sum of the interior angles derived by using Theorem 11.1. PROOF Use a paragraph proof to prove the Corollary to Theorem 11.2. See margin. TECHNOLOGY In Exercises 47 and 48, use geometry software to construct a polygon. At each vertex, extend one of the sides of the polygon to form an exterior angle. 46. 47. Measure each exterior angle and verify that the sum of the measures is 360°. Check results. 48. Move any vertex to change the shape of your polygon. What happens to the measures of the exterior angles? What happens to their sum? See margin. 49. HOUSES Pentagon ABCDE is an outline of the front of a house. Find the measure of each angle. 50. TENTS Heptagon PQRSTUV is an outline of a camping tent. Find the unknown angle measures. 46. Let A be a regular convex n-gon. Since all the m™A = m™E = 90°, m™B = m™C = m™P = m™V = 70°, m™S = 140° S interior √ are £ and each m™D = 120° C interior ™ and one of the 2xⴗ R T 150ⴗ 150ⴗ exterior √ at that vertex form a linear pair, the q 160ⴗ 160ⴗ U B D exterior √ are also £ by the Congruent xⴗ xⴗ Supplements Thm. Let x° P V A E be the measure of each exterior ™. Then n • x° = POSSIBLE POLYGONS Would it be possible for a regular polygon to have 360° interior angles with the angle measure described? Explain. 51–54. See margin. 360° and x° = }} 55. ƒ(n ) is the measure of each interior å of a regular n-gon; as n gets larger and larger, ƒ(n ) increases, becoming closer and closer to 180°. 56. ƒ(n ) is the measure of each exterior å of a regular n-gon; as n gets larger and larger, ƒ(n ) gets smaller and smaller. n. 51. 150° 52. 90° 53. 72° 54. 18° xy USING ALGEBRA In Exercises 55 and 56, you are given a function and its graph. In each function, n is the number of sides of a polygon and ƒ(n) is measured in degrees. How does the function relate to polygons? What happens to the value of ƒ(n) as n gets larger and larger? 55, 56. See margin. 48. If the measure of an 180n º 360 interior ™ increases, the 55. ƒ(n) = ᎏnᎏ measure of the corresp. exterior ™ decreases. If ƒ(n) the measure of an interior 120 ™ decreases, the measure 90 of the corresp. exterior ™ increases. The sum of the 60 measures of the exterior 30 √ is always 360°. 0 (n – 2) • 180° 51. Yes; if }} = 150°, n 0 360 56. ƒ(n) = ᎏᎏ n ƒ(n) 120 90 60 30 0 3 4 5 6 7 8 n 0 For Lesson 11.1: then n = 12. A regular 12LOGICAL REASONING You are gon (dodecagon) has inte- 57. rior √ with measure 150°. shown part of a convex n-gon. The 52. Yes; a square is a regular polygon with interior √ of 90°. pattern of congruent angles continues around the polygon. Use the Polygon Exterior Angles Theorem to find the value of n. 10 ADDITIONAL PRACTICE AND RETEACHING 3 4 5 6 7 8 n 163ⴗ 125ⴗ 11.1 Angle Measures in Polygons 667 • Practice Levels A, B, and C (Chapter 11 Resource Book, p. 14) • Reteaching with Practice (Chapter 11 Resource Book, p. 17) • See Lesson 11.1 of the Personal Student Tutor For more Mixed Review: • Search the Test and Practice Generator for key words or specific lessons. 667 Test Preparation 4 ASSESS QUANTITATIVE COMPARISON In Exercises 58–61, choose the statement that is true about the given quantities. A ¡ B ¡ C ¡ D ¡ DAILY HOMEWORK QUIZ Transparency Available 1. Find the sum of the measures of the interior angles in a convex 16-gon. 2520° 2. Find the value of x. 156° The two quantities are equal. The relationship cannot be determined from the given information. Column B 58. The sum of the interior angle measures of a decagon The sum of the interior angle measures of a 15-gon B 59. The sum of the exterior angle measures of an octagon 8(45°) C 60. m™1 m™2 45° 124° The quantity in column B is greater. Column A 125° x° The quantity in column A is greater. 118ⴗ 3. The measure of each interior angle of a regular polygon is 162°. How many sides does the polygon have? 20 4. The measure of each exterior angle of a regular polygon is 6°. How many sides does the polygon have? 60 1 61. ★ Challenge 135ⴗ 91ⴗ 156ⴗ 2 111ⴗ 146ⴗ Number of sides of a polygon with an exterior angle measuring 72° A 70ⴗ 72ⴗ D Number of sides of a polygon with an exterior angle measuring 144° Æ Æ 62. Polygon STUVWXYZ is a regular octagon. Suppose sides ST and UV are extended to meet at a point R. Find the measure of ™TRU. 90° EXTRA CHALLENGE NOTE Challenge problems for Lesson 11.1 are available in blackline format in the Chapter 11 Resource Book, p. 22 and at www.mcdougallittell.com. MIXED REVIEW FINDING AREA Find the area of the triangle described. (Review 1.7 for 11.2) 63. base: 11 inches; height: 5 inches 64. base: 43 meters; height: 11 meters 236.5 m2 27.5 in.2 65. vertices: A(2, 0), B(7, 0), C(5, 15) 66. vertices: D(º3, 3), E(3, 3), F(º7, 11) 37.5 sq. units 24 sq. units VERIFYING RIGHT TRIANGLES Tell whether the triangle is a right triangle. (Review 9.3) ADDITIONAL TEST PREPARATION 1. WRITING Explain how to find the sum of the measures of the interior angles of a polygon using triangles. Use diagonals to 67. no divide the polygon into triangular regions. The sum of the angles in each triangle is 180°. Multiply the number of triangles by 180° to get the sum of the measures of the interior angles in the polygon. 68. yes 69. no 75 21 16 13 7 5 72 2兹17 9 Æ Æ FINDING MEASUREMENTS GD and FH are diameters of circle C. Find the indicated arc measure. (Review 10.2) ៣ ៣ 72. mEH 70. mDH 80° 145° ៣ 65° ២ 245° 73. mEHG 71. mED G F 35ⴗ C 80ⴗ E D 668 668 Chapter 11 Area of Polygons and Circles H