L.T. I can identify special angle pairs and use
their relationships to find angle measure.
A. Vertical Angles
Previously, you learned that two angles are adjacent if they share a
common vertex and side but have no common interior points. In this
lesson, you will study other relationships between pairs of angles.
Two angles are vertical angles if
their sides form two pairs of
opposite rays.
1
2
4
3
1 and 3 are vertical
angles.
Vertical Angle
Pairs are
CONGRUENT
2 and 4 are vertical
angles.
B. Linear Pairs
Two adjacent angles are a linear pair if the form a
straight line.
30
5
°
6
150°
5 and 6 are a linear
pair.
Linear Angle Pairs
add up to 180°.
Identifying Vertical Angles and Linear Pairs
Answer the questions using the diagram.
Are 2 and 3 a linear pair?
1
2
4
3
SOLUTION
No. The angles are adjacent but their noncommon sides are not opposite
rays.
Identifying Vertical Angles and Linear Pairs
Answer the questions using the diagram.
Are 2 and 3 a linear pair?
Are 3 and 4 a linear pair?
1
2
4
3
SOLUTION
No. The angles are adjacent but their noncommon sides are not opposite
rays.
Yes. The angles are adjacent and their noncommon sides are
opposite rays.
Identifying Vertical Angles and Linear Pairs
Answer the questions using the diagram.
Are 2 and 3 a linear pair?
Are 3 and 4 a linear pair?
Are 1 and 3 vertical angles?
1
2
4
3
SOLUTION
No. The angles are adjacent but their noncommon sides are not opposite
rays.
Yes. The angles are adjacent and their noncommon sides are
opposite rays.
No. The sides of the angles do not form two pairs of opposite
rays.
Identifying Vertical Angles and Linear Pairs
Answer the questions using the diagram.
Are 2 and 3 a linear pair?
Are 3 and 4 a linear pair?
Are 1 and 3 vertical angles?
Are 2 and 4 vertical angles?
1
2
4
3
SOLUTION
No. The angles are adjacent but their noncommon sides are not opposite
rays.
Yes. The angles are adjacent and their noncommon sides are
opposite rays.
No. The sides of the angles do not form two pairs of opposite
rays.
No. The sides of the angles do not form two pairs of
opposite rays.
Finding Angle Measures
In the stair railing shown, 6 has a measure of 130˚. Find the
measures of the other three angles.
SOLUTION
6 and 8 are vertical angles. So, they are
congruent and have the same measure.
m 8 = m 6 = 130˚
6 and 7 are a linear pair. So, the sum of
their measures is 180˚.
m6 + m7 = 180˚
130˚ + m7 = 180˚
50
°
5
130°
13 8 6130°
13
0°
7 0°
50
°
m7 = 50˚
7 and 5 are vertical angles. So, they are
congruent and have the same measure.
All 4 angles together equal 360°
C. Supplementary Angles
Definition: Two angles are supplementary if the sum of
their measures is 180 degrees. Each angle is the
supplement of the other.
1
2
20
160
These are supplements of each other
because their angles add up to 180.
Example 1
Find the value of x.
x
20
x + 20 = 180
x = 160
Example 2
Find the value of x.
x
65
x + 65 = 180
x = 115
Example 3
Find the value of x.
(7x  10)
3x
(7x + 10) + 3x = 180
10x + 10 = 180
10x = 170
x = 17
D. Complementary Angles
Definition: Two angles are complementary if the sum of
their measures is 90 degrees. Each angle is the
complement of the other.
1
2
30
60
These are complements of each other because
their angles add up to be 90.
Example 4
Find the value of x.
x
x + 15 = 90
x = 75
15
Example 5
Find the value of x.
(4x + 3)
(x - 8)
(4x + 3) + (x - 8) = 90
5x - 5 = 90
5x = 95
x = 19
D. Angle Bisector
Definition: An angle bisector is a ray that divides an angle into two
congruent angles. It cuts the angle in half.