2.3 Complementary and

Transcription

1 2.3 omplementary and Supplementary ngles Goal Find measures of complementary and supplementary angles. ey Words complementary angles supplementary angles adjacent angles theorem Two angles are complementary angles if the sum of their measures is 90. Each angle is the complement of the other a and a are complementary angles. ma ma Two angles are supplementary angles if the sum of their measures is 180. Each angle is the supplement of the other a and a are supplementary angles. ma ma Visualize It! 1 2 a1 and a2 are complementary. a3 d a4 ppl nt y. omplementary angles make up the orner of a piece of paper. Supplementary angles make up the Side of a piece of paper. 3 4 EXMPE 1 Identify omplements and Supplements etermine whether the angles are complementary, supplementary, or neither. a. 22 b. c a. ecause , the angles are supplementary. b. ecause , the angles are neither complementary nor supplementary. c. ecause , the angles are complementary Ide tify o le ts d Su le ts etermine whether the angles are complementary, supplementary, or neither omplementary and Supplementary ngles 67

2 Student Help STUY TIP You can use numbers to refer to angles. Make sure that you do not confuse angle names with angle measures. Two angles are adjacent angles if they share a common vertex and side, but have no common interior points. common side a1 and a2 are adjacent angles. 1 2 common vertex EXMPE 2 Identify djacent ngles Tell whether the numbered angles are adjacent or nonadjacent. a. b. c a. ecause the angles do not share a common vertex or side, a1 and a2 are nonadjacent. b. ecause the angles share a common vertex and side, and they do not have any common interior points, a3 and a4 are adjacent. c. lthough a5 and a6 share a common vertex, they do not share a common side. Therefore, a5 and a6 are nonadjacent. EXMPE 3 Measures of omplements and Supplements a. a is a complement of a, and ma 47. Find ma. b. ap is a supplement of ar, and mar 36. Find map. a. a and a are complements, b. ap and ar are supplements, so their sum is 90. so their sum is 180. ma ma 90 map mar ma 90 map ma map ma 43 map 144 Measures of omplements and Supplements 4. a is a complement of a, and ma 79. Find ma. 5. ag is a supplement of ah, and mag 115. Find mah. 68 hapter 2 Segments and ngles

3 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 VISU STRTEGY raw examples of these theorems with specific measures, as shown on p ongruent omplements Theorem Words If two angles are complementary to the same angle, then they are congruent Symbols If ma1 ma2 90 and ma2 ma3 90, then a1 c a ongruent Supplements Theorem Words If two angles are supplementary to the same angle, then they are congruent Symbols If ma4 ma5 180 and ma5 ma6 180, then a4 c a6. You can use theorems in your reasoning about geometry, as shown in Example 4. EXMPE 4 Use a Theorem a7 and a8 are supplementary, and a8 and a9 are supplementary. Name a pair of congruent angles. Explain your reasoning a7 and a9 are both supplementary to a8. So, by the ongruent Supplements Theorem, a7 ca9. Use a Theorem 6. In the diagram, ma10 ma11 90, and ma11 ma Name a pair of congruent angles. Explain your reasoning omplementary and Supplementary ngles 69

4 2.3 Exercises Guided Practice Vocabulary heck Skill heck 1. Explain the difference between complementary angles and supplementary angles. 2. omplete the statement: Two angles are? if they share a common vertex and a common side, but have no common interior points. In Exercises 3 5, determine whether the angles are complementary, supplementary, or neither. lso tell whether the angles are adjacent or nonadjacent a is a complement of a, and ma 10. Find ma. 7. a is a supplement of a, and ma 109. Find ma. Practice and pplications Extra Practice See p Identifying ngles etermine whether the angles are complementary, supplementary, or neither. lso tell whether the angles are adjacent or nonadjacent Identifying ngles etermine whether the two angles shown on the clock faces are complementary, supplementary, or neither Homework H lp Example 1: Exs. 8 14, Example 2: Exs Example 3: Exs , 34 Example 4: Exs hapter 2 Segments and ngles

5 Finding omplements Find the measure of a complement of the angle given a is a complement of a, and ma 74. Find ma. 19. ap is a complement of aq, and map 9. Find maq. Finding Supplements Find the measure of a supplement of the angle given a is a supplement of a, and ma 96. Find ma. 24. ap is a supplement of aq, and map 7. Find maq. areers Finding omplements and Supplements Find the measures of a complement and a supplement of the angle. 25. ma ma ma ridges The lamillo ridge in Seville, Spain, was designed by Santiago alatrava. In the bridge, ma1 58, and ma2 24. Find the measures of the supplements of both a1 and a2. RHITET Santiago alatrava, a Spanish born architect, has developed designs for bridges, train stations, stadiums, and art museums. 1 2 areer inks SSZONE.OM Naming ngles In the diagram, aqpr is a right angle. 29. Name a straight angle. 30. Name two congruent supplementary angles. R S 31. Name two supplementary angles that are not congruent. P P T 32. Name two complementary angles. 2.3 omplementary and Supplementary ngles 71

6 each hairs djustable beach chairs form angles that are supplementary. Find the value of x x 140 x Student Help I SSZONE.OM HOMEWOR HEP Extra help with problem solving in Exs is at classzone.com Using lgebra a and a are complementary angles. Find the value of the variable x 13x 8n 7n (3k 10) 2k 38. omplementary ngles a and ae are complements, and ae and ae are complements. an you show that a cae? Explain. E 39. Technology Use geometry software to draw two intersecting lines. Measure three of the four angles formed. rag the points and observe the angle measures. What theorem does this illustrate? P omplements and Supplements Find the angle measure described. 40. a1 and a2 are both supplementary to a3, and ma1 43. Find the measure of a a4 and a6 are both complementary to a5, and ma5 85. Find the measure of a ap is supplementary to aq, ar is supplementary to ap, and maq 60. Find the measure of ar. 43. hallenge a and a are supplementary angles. The measure of a is eight times the measure of a. Find ma and ma. 72 hapter 2 Segments and ngles

7 Standardized Test Practice 44. Multiple hoice What is the measure of a complement of a 27 angle? Multiple hoice a1 and a2 are supplementary. Suppose that ma1 60 and ma2 (2x 20). What is the value of x? F 5 G 10 H 50 J 100 Mixed Review Segment ddition Postulate Find the length. (esson 1.5) 46. Find FH. 47. Find. F 4.5 G 8.2 H J Midpoint Formula Find the coordinates of the midpoint of &*. (esson 2.1) 48. (0, 0), (8, 2) 49. ( 6, 0), (2, 4) 50. (4, 1), (10, 3) 51. ( 2, 5), ( 2, 7) 52. (3, 8), ( 1, 0) 53. ( 5, 9), (11, 5) lgebra Skills Evaluating ecimals Evaluate. (Skills Review, p. 655) Quiz 1 1. In the diagram, is the midpoint of J&. Find and J. (esson 2.1) J 17 Find the coordinates of the midpoint of &*. (esson 2.1) 2. (1, 3), (7, 1) 3. ( 4, 2), (6, 4) 4. ( 5, 3), (3, 3) In Exercises 5 7, M&*( bisects aj. Find the angle measure. (esson 2.2) 5. Find majm. 6. Find maj. 7. Find maj. J 82 M J M 11 J 58 M 8. af is a supplement of ag, and maf 101. Find mag. (esson 2.3) 9. The measure of a is 83. Find the measure of a complement and a supplement of a. (esson 2.3) 2.3 omplementary and Supplementary ngles 73