Complementary and Supplementary Angles

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This pdf includes the following topics:-
Identify Complements and Supplements
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
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
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
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
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.
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
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
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
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