This pdf includes the following topics:- Identify Complements and Supplements Adjacent angles Identify Adjacent Angles Measures of Complements and Supplements Congruent Complements Theorem Congruent Supplements Theorem Exercises
1. Page 1 of 7 2.3 Complementary and Supplementary Angles Goal Two angles are complementary angles if the sum of their measures Find measures of is 908. Each angle is the complement of the other. complementary and supplementary angles. A aA and aB are complementary angles. Key Words 328 588 maA 1 maB 5 328 1 588 5 908 • complementary angles B • supplementary angles • adjacent angles Two angles are supplementary angles if the sum of their measures is • theorem 1808. Each angle is the supplement of the other. 1348 aC and aD are supplementary angles. 468 maC 1 maD 5 1348 1 468 5 1808 C D EXAMPLE 1 Identify Complements and Supplements Visualize It! Determine whether the angles are complementary, supplementary, or neither. 2 a1 and a2 are 1 complementary. a. 228 b. c. 1588 3 158 858 558 a3 and a4 are 358 supplementary. 4 Solution a. Because 228 1 1588 5 1808, the angles are supplementary. Complementary angles make up the Corner of b. Because 158 1 858 5 1008, the angles are neither complementary a piece of paper. Supplementary angles nor supplementary. make up the Side of a c. Because 558 1 358 5 908, the angles are complementary. piece of paper. Identify Complements and Supplements Determine whether the angles are complementary, supplementary, or neither. 1. 2. 3. 308 1488 418 398 498 328 2.3 Complementary and Supplementary Angles 67
2. Page 2 of 7 Student Help Two angles are adjacent angles if they share a common vertex and side, but have no common interior points. STUDY TIP You can use numbers to refer to angles. common side Make sure that you do not confuse angle a1 and a2 are adjacent angles. names with angle 1 2 measures. common vertex EXAMPLE 2 Identify Adjacent Angles Tell whether the numbered angles are adjacent or nonadjacent. a. b. c. 2 1 5 3 6 4 Solution a. Because the angles do not share a common vertex or side, a1 and a2 are nonadjacent. b. Because the angles share a common vertex and side, and they do not have any common interior points, a3 and a4 are adjacent. c. Although a5 and a6 share a common vertex, they do not share a common side. Therefore, a5 and a6 are nonadjacent. EXAMPLE 3 Measures of Complements and Supplements a. aA is a complement of aC, and maA 5 478. Find maC. b. aP is a supplement of aR, and maR 5 368. Find maP. Solution a. aA and aC are complements, b. aP and aR are supplements, so their sum is 908. so their sum is 1808. maA 1 maC 5 908 maP 1 maR 5 1808 478 1 maC 5 908 maP 1 368 5 1808 478 1 maC 2 478 5 908 2 478 maP 1 368 2 368 5 1808 2 368 maC 5 438 maP 5 1448 Measures of Complements and Supplements 4. aB is a complement of aD, and maD 5 798. Find maB. 5. aG is a supplement of aH, and maG 5 1158. Find maH. 68 Chapter 2 Segments and Angles
3. Page 3 of 7 A theorem is a true statement that follows from other true statements. The two theorems that follow are about complementary and supplementary angles. Student Help THEOREMS 2.1 and 2.2 VISUAL STRATEGY Draw examples of 2.1 Congruent Complements Theorem these theorems with Words If two angles are complementary specific measures, as to the same angle, then they are shown on p. 52. 3 congruent. 2 1 Symbols If ma1 1 ma2 5 908 and ma2 1 ma3 5 908, then a1 c a3. 2.2 Congruent Supplements Theorem Words If two angles are supplementary 5 to the same angle, then they are congruent. 4 6 Symbols If ma4 1 ma5 5 1808 and ma5 1 ma6 5 1808, then a4 c a6. You can use theorems in your reasoning about geometry, as shown in Example 4. EXAMPLE 4 Use a Theorem a7 and a8 are supplementary, and a8 and 8 a9 are supplementary. Name a pair of 7 9 congruent angles. Explain your reasoning. Solution a7 and a9 are both supplementary to a8. So, by the Congruent Supplements Theorem, a7 c a9. Use a Theorem 6. In the diagram, ma10 1 ma11 5 908, and ma11 1 ma12 5 908. Name a pair of congruent angles. Explain your reasoning. 10 11 12 2.3 Complementary and Supplementary Angles 69
4. Page 4 of 7 2.3 Exercises Guided Practice Vocabulary Check 1. Explain the difference between complementary angles and supplementary angles. 2. Complete the statement: Two angles are __?__ if they share a common vertex and a common side, but have no common interior points. Skill Check In Exercises 3–5, determine whether the angles are complementary, supplementary, or neither. Also tell whether the angles are adjacent or nonadjacent. 3. 4. 908 5. 308 758 1108 1508 158 6. aA is a complement of aB, and maA 5 108. Find maB. 7. aC is a supplement of aD, and maD 5 1098. Find maC. Practice and Applications Extra Practice Identifying Angles Determine whether the angles are See p. 677. complementary, supplementary, or neither. Also tell whether the angles are adjacent or nonadjacent. 8. 9. 10. 678 588 318 788 1028 338 Identifying Angles Determine whether the two angles shown on the clock faces are complementary, supplementary, or neither. 11. 12. Homework Help Example 1: Exs. 8–14, 30–32 13. 14. Example 2: Exs. 8–10 Example 3: Exs. 15–28 33, 34 Example 4: Exs. 38–42 70 Chapter 2 Segments and Angles
5. Page 5 of 7 Finding Complements Find the measure of a complement of the angle given. 15. 16. 17. 868 248 418 18. aK is a complement of aL, and maK 5 748. Find maL. 19. aP is a complement of aQ, and maP 5 98. Find maQ. Finding Supplements Find the measure of a supplement of the angle given. 20. 21. 22. 558 1608 148 23. aA is a supplement of aB, and maA 5 968. Find maB. 24. aP is a supplement of aQ, and maP 5 78. Find maQ. Finding Complements and Supplements Find the measures of a complement and a supplement of the angle. 25. maA 5 398 26. maB 5 898 27. maC 5 548 Careers 28. Bridges The Alamillo Bridge in Seville, Spain, was designed by Santiago Calatrava. In the bridge, ma1 5 588, and ma2 5 248. Find the measures of the supplements of both a1 and a2. ARCHITECT Santiago Calatrava, a Spanish born architect, has developed 1 2 designs for bridges, train stations, stadiums, and art Career Links Naming Angles In the diagram, aQPR is a right angle. CLASSZONE.COM 29. Name a straight angle. R S 30. Name two congruent supplementary angles. 31. Name two supplementary angles that are not congruent. P P T 32. Name two complementary angles. 2.3 Complementary and Supplementary Angles 71
6. Page 6 of 7 Beach Chairs Adjustable beach chairs form angles that are supplementary. Find the value of x. 33. 34. 1168 x8 1408 x8 IStudent Help Using Algebra aABD and aDBC are complementary angles. Find ICLASSZONE.COM the value of the variable. HOMEWORK HELP 35. 36. 37. A B Extra help with problem A (3k 1 10)8 C D 5x 8 D solving in Exs. 35–37 is 13x 8 at classzone.com 8n8 7n8 D 2k8 C B C A B 38. Complementary Angles aABD and aDBE D are complements, and aCBE and aDBE E are complements. Can you show that aABD c aCBE? Explain. A B C 39. Technology Use geometry software to draw two intersecting lines. Measure three of the four A C angles formed. Drag the points P and observe the angle measures. D What theorem does this illustrate? B Complements and Supplements Find the angle measure described. 40. a1 and a2 are both supplementary to a3, and ma1 5 438. Find the measure of a2. 41. a4 and a6 are both complementary to a5, and ma5 5 858. Find the measure of a4. 42. aP is supplementary to aQ, aR is supplementary to aP, and maQ 5 608. Find the measure of aR. 43. Challenge aC and aD are supplementary angles. The measure of aD is eight times the measure of aC. Find maC and maD. 72 Chapter 2 Segments and Angles
7. Page 7 of 7 Standardized Test 44. Multiple Choice What is the measure of a complement of a Practice 278 angle? X A 538 B 638 X C 1178 X D 1638 X 45. Multiple Choice a1 and a2 are supplementary. Suppose that ma1 5 608 and ma2 5 (2x 1 20)8. What is the value of x? X F 5 G 10 X H 50 X J 100 X Mixed Review Segment Addition Postulate Find the length. (Lesson 1.5) 46. Find FH. 47. Find KL. 25 F 4.5 G 8.2 H J 13 K L &*. Midpoint Formula Find the coordinates of the midpoint of AB (Lesson 2.1) 48. A(0, 0), B(8, 2) 49. A(26, 0), B(2, 4) 50. A(4, 1), B(10, 3) 51. A(22, 5), B(22, 7) 52. A(3, 28), B(21, 0) 53. A(25, 29), B(11, 5) Algebra Skills Evaluating Decimals Evaluate. (Skills Review, p. 655) 54. 2.58 1 8.04 55. 5.17 2 1.96 56. 1.4 3 3.1 57. 0.61 3 0.38 58. 11.2 4 1.4 59. 2 3 5.4 3 3.9 Quiz 1 &. 1. In the diagram, K is the midpoint of JL Find KL and JL. (Lesson 2.1) J 17 K L &*. (Lesson 2.1) Find the coordinates of the midpoint of AB 2. A(1, 3), B(7, 21) 3. A(24, 22), B(6, 4) 4. A(25, 3), B(3, 23) &*( bisects aJKL. Find the angle measure. (Lesson 2.2) In Exercises 5–7, KM 5. Find maJKM. 6. Find maJKL. 7. Find maJKL. K K L J M 588 J 118 J 828 M L K L M 8. aF is a supplement of aG, and maF 5 1018. Find maG. (Lesson 2.3) 9. The measure of aD is 838. Find the measure of a complement and a supplement of aD. (Lesson 2.3) 2.3 Complementary and Supplementary Angles 73
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