Lesson 1-5
Pairs of
Angles
Lesson 1-5: Pairs of Angles
1
Adjacent Angles
Definition: A pair of angles with a shared vertex and common
side but do not have overlapping interiors.
Examples: 1 and 2 are adjacent. 3 and 4 are not.
1 and ADC are not adjacent.
A
B
4
36°
22°
1
3
C
2
D
Adjacent Angles( a common side )
Non-Adjacent Angles
Lesson 1-5: Pairs of Angles
2
Complementary Angles
Definition: A pair of angles whose sum is 90˚
m2 = 50°
Examples:
A
B
2
Q
A
B
F
2
1
C
Q
1
R
Adjacent Angles
( a common side )
m1 = 40°
G
Non-Adjacent Angles
Lesson 1-5: Pairs of Angles
3
Supplementary Angles
Definition: A pair of angles whose sum is 180˚
Examples:
B
Adjacent supplementary angles are
also called “Linear Pair.”
2
1
Q
A
C
B
F
Non-Adjacent Angles
2
m1 = 40°
m2 = 140°
A
1
Q
R
Lesson 1-5: Pairs of Angles
G
4
Vertical Angles
Definition: A pair of angles whose sides form opposite rays.
Examples:
A
1
1 and 3
4
2 and 4
B
D
Q
2
3
C
Vertical angles are non-adjacent angles formed by intersecting
lines.
Lesson 1-5: Pairs of Angles
5
Theorem: Vertical Angles are =~
Given:
The diagram
Prove:
1  3
A
1
4
D
Statements
1. m1 + m2 = 180°
B
Q
2
3
Reasons
m2 + m3 = 180°
2. m1 + m2 = m2 + m3
C
1. Definition: Linear Pair
2. Property: Substitution
3. m1 = m3
3. Property: Subtraction
4. m1  m3
4. Definition: Congruence
Lesson 1-5: Pairs of Angles
6
What’s “Important” in Geometry?
4 things to always look for !
180˚
360˚
. . . and
Congruence
90˚
Most of the rules (theorems)
and vocabulary of Geometry
are based on these 4 things.
Lesson 1-5: Pairs of Angles
7
Example: If m4 = 67º, find the measures
of all other angles.
Step 1: Mark the figure with given info.
Step 2: Write an equation.
m3  m4  180
67º
m3  67 180
3
4
2
1
m3 180  67  113
Because 4 and 2 arevertical angles, they are equal. m4  m2  67
Because 3 and 1 are vertical angles, they are equal. m3  m1  117
Lesson 1-5: Pairs of Angles
8
Example: If m1 = 23 º and m2 = 32 º, find the
measures of all other angles.
Answers:
m4  23 (1 & 4 are vertical angles.)
m5  32 (2 & 5 are vertical angles.)
m1  m2  m3  180
2
23  32  m3  180
m3  180  55  125
1
m3  m6  125
3
6
4
5
3 & 6 are vertical angles.
Lesson 1-5: Pairs of Angles
9
Example: If m1 = 44º, m7 = 65º find the
measures of all other angles.
Answers: m3  90
m1  m4  44
m4  m5  90
44  m5  90
m5  46
4
5
6
3
2
1
7
m6  m7  90
m6  65  90
m6  25
Lesson 1-5: Pairs of Angles
10
Algebra and Geometry
Common Algebraic Equations used in Geometry:
(
(
(
(
)+(
)+(
)+(
)=( )
)=( )
) = 90˚
) = 180˚
If the problem you’re working on has a variable (x),
then consider using one of these equations.
Lesson 1-5: Pairs of Angles
11