Download 2.4 Vertical Angles

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2.4
Goal
Find the measures of
angles formed by
intersecting lines.
Vertical Angles
Two angles are vertical angles if they are not adjacent and their
sides are formed by two intersecting lines. The scissors show
two sets of vertical angles.
2
a1 and a3 are vertical angles.
Key Words
1
a2 and a4 are vertical angles.
3
4
• vertical angles
• linear pair
Two adjacent angles are a linear pair if their
noncommon sides are on the same line.
common
side
a5 and a6 are a linear pair.
5
6
noncommon side
noncommon side
Identify Vertical Angles and Linear Pairs
Visualize It!
EXAMPLE
You can use colored
pencils to help you see
pairs of vertical angles.
Determine whether the labeled angles are vertical angles,
a linear pair, or neither.
1
a.
1
6
2
b.
3
4
1
2
c.
3
5
4
6
5
Vertical angles
a1 and a4
a2 and a5
a3 and a6
Solution
a. a1 and a2 are a linear pair because they are adjacent and their
noncommon sides are on the same line.
b. a3 and a4 are neither vertical angles nor a linear pair.
c. a5 and a6 are vertical angles because they are not adjacent and
their sides are formed by two intersecting lines.
POSTULATE 7
Linear Pair Postulate
Words
If two angles form a linear pair,
then they are supplementary.
Symbols
1
2
ma1 ma2 180
2.4
Vertical Angles
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EXAMPLE
2
Use the Linear Pair Postulate
Find the measure of aRSU.
U
62ⴗ
R
Solution
S
T
aRSU and aUST are a linear pair. By the Linear Pair Postulate, they
are supplementary. To find maRSU, subtract maUST from 180.
maRSU 180 maUST 180 62 118
Student Help
THEOREM 2.3
VISUAL STRATEGY
Vertical Angles Theorem
Draw an example of
this theorem with
specific measures,
as shown on p. 52.
Words
Vertical angles are congruent.
Symbols
1
a1 c a3 and a2 c a4.
2
3
4
The following steps show why the Vertical Angles Theorem is true.
1
●
a1 and a2 are a linear pair, so a1 and a2
are supplementary.
2
●
a2 and a3 are a linear pair, so a2 and a3
are supplementary.
1
3
●
a1 and a3 are supplementary to the same
angle, so a1 is congruent to a3 by the
Congruent Supplements Theorem.
1
1
2
2
3
3
Student Help
LOOK BACK
To review the Congruent
Supplements Theorem,
see p. 69.
EXAMPLE
3
2
3
Use the Vertical Angles Theorem
Find the measure of aCED.
D
A
E
50ⴗ
Solution
C
B
aAEB and aCED are vertical angles. By the Vertical Angles Theorem,
aCED c aAEB, so maCED maAEB 50.
76
Chapter 2
Segments and Angles
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Student Help
STUDY TIP
When you know the
measure of one vertical
angle, an easy starting
point is to fill in the
measure of the other.
EXAMPLE
Find Angle Measures
4
Find ma1, ma2, and ma3.
35ⴗ
1
3
2
Solution
ma2 35
Vertical Angles Theorem
ma1 180 35ⴗ 145
Linear Pair Postulate
ma3 ma1 145
Vertical Angles Theorem
Find Angle Measures
Find ma1, ma2, and ma3.
1.
2.
1
28ⴗ
3.
2
1
3
EXAMPLE
5
2
124ⴗ
67ⴗ
3
2 1
3
Use Algebra with Vertical Angles
Find the value of y.
(4y ⴚ 42)ⴗ
2y ⴗ
Solution
Because the two expressions are measures of vertical angles, you can
write the following equation.
(4y 42) 2y
Vertical Angles Theorem
4y 42 4y 2y 4y
Subtract 4y from each side.
42 2y
Simplify.
42
2y
2
2
Divide each side by 2.
21 y
Simplify.
Use Algebra with Angle Measures
Find the value of the variable.
4.
5.
(2r ⴙ 3)ⴗ
6.
(20t ⴙ 5)ⴗ
3xⴗ
89ⴗ
(2x ⴙ 16)ⴗ
15t ⴗ
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Vertical Angles
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2.4 Exercises
Guided Practice
Vocabulary Check
Complete the statement.
1. Two adjacent angles whose noncommon sides are on the same
line are called __?__.
2. Two angles are called __?__ if they are not adjacent and their sides
are formed by two intersecting lines.
Skill Check
Find the measure of the numbered angle.
3.
4.
1
2
19ⴗ
19ⴗ
Find ma1, ma2, and ma3.
5.
6.
72ⴗ 1
3 2
1
2
3
Find the value of x.
7.
8.
(12x ⴚ 8)ⴗ
40ⴗ
6x ⴗ 12x ⴗ
Practice and Applications
Extra Practice
Vertical Angles and Linear Pairs Determine whether the angles are
vertical angles, a linear pair, or neither.
See p. 678.
Homework Help
Example 1:
Example 2:
Example 3:
Example 4:
Example 5:
78
Exs. 9–14
Exs. 15–19
Exs. 20–22
Exs. 28–37
Exs. 51–56
Chapter 2
9. a5 and a6
10. a5 and a9
11. a5 and a8
12. a6 and a9
13. a8 and a9
14. a5 and a7
5
Using the Linear Pair Postulate Find the measure of a1.
15.
Segments and Angles
16.
1
37ⴗ
17.
1
91ⴗ
1
136ⴗ
6 7
9 8
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Linear Pairs Find the measure of the angle described.
18. a1 and a2 are a linear pair, and ma1 51. Find ma2.
19. a3 and a4 are a linear pair, and ma4 124. Find ma3.
Using the Vertical Angles Theorem Find the measure of a1.
20.
21.
22.
160ⴗ
1
1
76ⴗ
1
63ⴗ
Evaluating Statements Use the figure below to decide whether the
statement is true or false.
23. If ma1 40, then ma2 140.
24. If ma4 130, then ma2 50.
1
25. a1 and a4 are a linear pair.
4
3
26. ma1 ma4 ma3 ma2
2
27. a1 and a4 are vertical angles.
IStudent Help
ICLASSZONE.COM
Finding Angle Measures Find ma1, ma2, and ma3.
28.
HOMEWORK HELP
Extra help with problem
solving in Exs. 28–33 is
at classzone.com
29.
1
3
25ⴗ
82ⴗ 1
3 2
2
31.
30.
2 3
1 117ⴗ
32.
3
2
55ⴗ
1
33.
3
12
121ⴗ 3
140ⴗ
2 1
Flags Each flag shown contains vertical angles. Find ma1, ma2,
and ma3.
34.
35.
1
2
61.9ⴗ
3
Scotland
36.
90ⴗ
1
3
2
2
1
3
126.9ⴗ
Dominican Republic
Jamaica
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Vertical Angles
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37. Drafting Table The legs of the drafting table form vertical angles.
Find the measures of a1, a2, and a3.
Careers
3
85ⴗ
2
1
Finding Angle Measures Find ma1, ma2, ma3, and ma4.
ERGONOMISTS study work
conditions to improve the
safety, efficiency, and comfort
of workers. Drafting tables are
angled so people can work at
them without injuring their
backs.
Career Links
CLASSZONE.COM
38.
39.
60ⴗ
4
40.
4
1 2
3
55ⴗ
3
2
44ⴗ
3
4
2
106ⴗ
1
1 38ⴗ
Vertical Angles Use the diagram to complete the statement.
41. aBGC c __?__
42. aAGB c __?__
43. aAGC c __?__
44. aCGE c __?__
45. maAGF __?__
46. maDGE __?__
47. maCGE __?__
48. maBGC __?__
49. maDGF __?__
50. maAGD __?__
B
C
50ⴗ G
22ⴗ
D
A
F
E
Using Algebra Find the value of the variable.
51.
52.
152ⴗ
53.
(w ⴙ 73)ⴗ
78ⴗ (5y ⴙ 2)ⴗ
105ⴗ (2x ⴚ 11)ⴗ
Using Algebra Find the value of the variable. Then use
substitution to find maABC.
54.
55.
A
E
A
B
8nⴗ
C
80
Chapter 2
Segments and Angles
D
(9n ⴚ 5)ⴗ
E
56.
(6x ⴙ 19)ⴗ C
xⴗ
B
D
C (5p ⴙ 28)ⴗ D
B
A
12pⴗ
E
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57. Challenge Find the values of x and y in the diagram below.
18y ⴗ
(7x ⴚ 2)ⴗ
58.
Standardized Test
Practice
(11x ⴚ 34)ⴗ
Visualize It! Sketch and label four angles so that a1 and a2
are acute vertical angles, a3 is a right angle adjacent to a2, and
a1 and a4 form a linear pair.
59. Multi-Step Problem Use the diagram below.
(4x ⴙ 8)ⴗ
6x ⴗ
a. Use the Vertical Angles Theorem to write an equation.
b. Solve your equation to find the value of x.
c. Find the measures of the acute angles formed by the lines.
d. Find the measures of the obtuse angles formed by the lines.
Mixed Review
Describing Number Patterns Describe a pattern in the numbers.
Write the next number you expect in the pattern. (Lesson 1.1)
60. 4, 11, 18, 25, . . .
61. 3, 15, 75, 375, . . .
62. 32, 16, 8, 4, . . .
63. 404, 414, 424, 434, . . .
Congruent Segments Determine which segments in the coordinate
plane are congruent. (Lesson 1.5)
64.
65.
y
E(4, 3)
A(4, 3)
C (2, 1)
B (4, 0)
G (3, 2)
Algebra Skills
2
J(3, 3)
D(1, 1)
1
y
L(2, 1)
2
M(1, 1)
N(3, 0)
x
F (4, 1)
H(1, 2)
K (2, 3)
Q (3, 2)
4 x
R (0, 2)
P(3, 3)
Simplifying Expressions Simplify the expression.
(Skills Review, p. 671)
66. 16x 9x
67. 7 a 2a
68. 8z 2 5z 2 4
69. 6b 2 6b b 2
70. 4(t 3) 4t
71. 3w 2 1 w 2 5
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