The Polygon Angle-Sum 6-1 Theorems Vocabulary Review 1. Underline the correct word to complete the sentence. In a convex polygon, no point on the lines containing the sides of the polygon is in the interior / exterior of the polygon. 2. Cross out the polygon that is NOT convex. Vocabulary Builder regular polygon (noun) REG yuh lur PAHL ih gahn Definition: A regular polygon is a polygon that is both equilateral and equiangular. d. e v Example: An equilateral triangle is a regular polygon with three congruent sides er s e and three congruent angles. s R ht g Ri All Use Your Vocabulary s. e at Underline the correct word(s) to complete each sentence. affili s 3. The sides of a regular polygon are congruent / scalene . or it 4. A right triangle is / is not a regular polygon. nc. n, I o 5. An isosceles triangle is / is not always a regular polygon. ati c u Write equiangular, equilateral, or regular to identify each hexagon. Use each d E n word once. o s ar e 6. 7. 120(cid:237) 8. 120(cid:237) 120(cid:237) y P b 120(cid:237) 120(cid:237) © 120(cid:237) 120(cid:237) ht g 120(cid:237) 120(cid:237) yri 120(cid:237) 120(cid:237) 120(cid:237) op C Chapter 6 146 Theorem 6-1 Polygon Angle-Sum Theorem and Corollary Theorem 6-1 Th e sum of the measures of the interior angles of an n-gon is (n 2 2)180. (n 2 2)180 Corollary Th e measure of each interior angle of a regular n-gon is . n 9. When n 2 2 5 1, the polygon is a(n) 9. 10. When n 2 2 5 2, the polygon is a(n) 9. Problem 1 Finding a Polygon Angle Sum Got It? What is the sum of the interior angle measures of a 17-gon? 11. Use the justifications below to find the sum. 12. Draw diagonals from vertex A to check your answer. sum 5 2 2 180 Polygon Angle-Sum Theorem A 5 Q 2 2R180 Substitute for n. 5 Q ? 180 R Subtract. 5 Simplify. 13. The sum of the interior angle measures of a 17-gon is . d. e v Problem 2 Using the Polygon Angle-Sum Theorem er s e R hts Got It? What is the measure of each interior angle in a regular nonagon? g Ri All Underline the correct word or number to complete each sentence. s. e 14. The interior angles in a regular polygon are congruent / different . at affili 15. A regular nonagon has 7 / 8 / 9 congruent sides. s or it 16. Use the Corollary to the Polygon Angle-Sum Theorem to find the measure of each nc. interior angle in a regular nonagon. n, I o ati 2 2 180 uc Measure of an angle 5 d n E Q R o s ar e y P 180 b © 5 ht Q R g yri p o 5 C 17. The measure of each interior angle in a regular nonagon is . 147 Lesson 6-1 Problem 3 Using the Polygon Angle-Sum Theorem Got It? What is mlG in quadrilateral EFGH? G 18. Use the Polygon Angle-Sum Theorem to find m/G for n 5 4. F 120(cid:24) m/E 1 m/F 1 m/G 1 m/H 5 (n 2 2)180 85(cid:24) 53(cid:24) H E m/E 1 m/F 1 m/G 1 m/H 5 2 2 180 1 1 1 5 Q ? 180R m/G 1 5 m/G 5 19. m/G in quadrilateral EFGH is . Theorem 6-2 Polygon Exterior Angle-Sum Theorem Th e sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. 20. In the pentagon below, m/1 1 m/2 1 m/3 1 m/4 1 m/5 5 . 3 2 4 1 5 Use the Polygon Exterior Angle-Sum Theorem to find each measure. 21. 120(cid:237) 22. 56(cid:237) 81(cid:237) 90(cid:237) ed. v 75(cid:237) er s e R 87(cid:237) 72(cid:237) 66(cid:237) hts 73(cid:237) g Ri All 120 1 81 1 1 87 5 360 90 1 1 75 1 73 1 66 5 s. e at affili s or it Problem 4 Finding an Exterior Angle Measure c. n n, I o Got It? What is the measure of an exterior angle of a regular nonagon? ati c u d Underline the correct number or word to complete each sentence. E n o s 23. Since the nonagon is regular, its interior angles are congruent / right . ar e P y b 24. The exterior angles are complements / supplements of the interior angles. © ht 25. Since the nonagon is regular, its exterior angles are congruent / right . yrig p o C 26. The sum of the measures of the exterior angles of a polygon is 180 / 360 . 27. A regular nonagon has 7 / 9 / 12 sides. Chapter 6 148 28. What is the measure of an exterior angle of a regular nonagon? Explain. _______________________________________________________________________ _______________________________________________________________________ Lesson Check • Do you UNDERSTAND? Error Analysis Your friend says that she measured an interior angle of a regular polygon as 130. Explain why this result is impossible. 29. Use indirect reasoning to find a contradiction. Assume temporarily that a regular n-gon has a 1308 interior angle. angle sum 5 ? n A regular n-gon has n congruent angles. angle sum 5 180 Polygon Angle-Sum Theorem 5 Q R180 Use the Transitive Property of Equality. 5 Q 2R Use the Distributive Property. 5 Subtract 180n from each side. n 5 Divide each side by 250. n 2 The number of sides in a polygon is a whole number $ 3. 30. Explain why your friend’s result is impossible. d. e v _______________________________________________________________________ er s e R hts _______________________________________________________________________ g Ri All _______________________________________________________________________ s. e at affili _______________________________________________________________________ s or it c. n n, I o ati Math Success c u d E on Check off the vocabulary words that you understand. s ar e P equilateral polygon equiangular polygon regular polygon y b © Rate how well you can fi nd angle measures of polygons. ht g yri p Need to 0 2 4 6 8 10 Now I o C review get it! 149 Lesson 6-1 Name Class Date Additional Problems 6-1 The Polygon Angle-Sum Theorems Problem 1 What is the sum of the angle measures of a 10-gon? Problem 2 Marcy creates a fl oor tile pattern using squares, regular hexagons, and regular dodecagons (12-sided polygons). What is the measure of each angle in one regular dodecagon? Prentice Hall Geometry • Additional Problems Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 69 Name Class Date Additional Problems 6-1 (continued) The Polygon Angle-Sum Theorems Problem 3 What is m/D in quadrilateral ABCD? A 98(cid:24) B 91(cid:24) 112(cid:24) C D Problem 4 What is the measure of an exterior angle 1 of a regular hexagon? Prentice Hall Geometry • Additional Problems Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 70 Name Class Date Practice 6-1 Form G The Polygon Angle-Sum Theorems Find the sum of the angle measures of each polygon. 1. 2. 3. 4. 12-gon 5. 18-gon 6. 25-gon 7. 60-gon 8. 102-gon 9. 17-gon 10. 36-gon 11. 90-gon 12. 11-gon Find the measure of one angle in each regular polygon. Round to the nearest tenth if necessary. 13. 14. 15. 16. regular 15-gon 17. regular 11-gon 18. regular 13-gon 19. regular 24-gon 20. regular 360-gon 21. regular 18-gon 22. regular 36-gon 23. regular 72-gon 24. regular 144-gon Algebra Find the missing angle measures. 25. 26. 27. 28. 29. Prentice Hall Gold Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 3 Name Class Date Practice (continued) Form G 6-1 The Polygon Angle-Sum Theorems Algebra Find the missing angle measures. 30. 31. 32. 33. 34. 35. Find the measure of an exterior angle of each regular polygon. Round to the nearest tenth if necessary. 36. decagon 37. 16-gon 38. hexagon 39. 20-gon 40. 72-gon 41. square 42. 15-gon 43. 25-gon 44. 80-gon Find the values of the variables for each regular polygon. Round to the nearest tenth if necessary. 45. 46. 47. 48. Reasoning Can a quadrilateral have no obtuse angles? Explain. The measure of an exterior angle of a regular polygon is given. Find the measure of an interior angle. Then find the number of sides. 49. 12 50. 6 51. 45 52. 40 53. 24 54. 18 55. 9 56. 14.4 57. 7.2 Prentice Hall Gold Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 4 Name Class Date Practice 6-1 Form K The Polygon Angle-Sum Theorems Find the sum of the angle measures of each polygon. 1. To start, determine the sum of the angles using the Polygon Angle-Sum Theorem. Sum = (n − 2)180 = ( – 2)180 = 2. 21-gon 3. 42-gon 4. 50-gon 5. 205-gon Find the measure of one angle in each regular polygon. 6. To start, write the formula used to calculate the measure of an angle of a regular polygon. Then substitute n = 9 into the formula. (n−2)180 ( −2)180 = = n . 8. Find the missing angle measures. 9. To start, determine the sum of the angles using the Polygon Angle-Sum Theorem. Sum = (n − 2)180 = (6 − 2)180 = 720 Write an equation relating each interior angle to the sum of the angles. n + 156 + 122 + 143 + 108 + 110 = 720 10. 11. Prentice Hall Foundations Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 5 Name Class Date Practice (continued) Form K 6-1 The Polygon Angle-Sum Theorems Find the measure of an exterior angle of each regular polygon. 12. 12-gon 13. 24-gon 14. 45-gon The sum of the angle measures of a polygon with n sides is given. Find n. 15. 900 16. 1440 17. 2340 18. Carly built a Ferris wheel using her construction toys. The frame of the wheel is a regular 16-gon. Find the sum of the angle measures of the Ferris wheel and the measure of one angle. Algebra Find the value of each variable. 19. 20. 21. Your friend wants to build the picture frame shown at the right. a. What regular polygon is the inside of the frame? b. Find the measure of each numbered angle. c. Reasoning If you extended one of the exterior sides of the outside of the frame, would the measure of the exterior angle be the same as the measure of ∠2? Explain. 22. Caning chair seats first became popular in England in the 1600s. This method of weaving natural materials produces a pattern that contains several polygons. Identify the outlined polygon. Then, assuming that the polygon is regular, find the measure of each numbered angle. 23. Algebra The measure of an interior angle of a regular polygon is four times the measure of an exterior angle of the same polygon. What is the name of the polygon? 24. Reasoning The measure of the exterior angle of a regular polygon is 30. What is the measure of an interior angle of the same polygon? Explain. Prentice Hall Foundations Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 6