vertical angles complementary angles

1. Lesson 1-5: Pairs of Angles 1
Lesson 1-5
Pairs of
Angles

2. Lesson 1-5: Pairs of Angles 2
Adjacent Angles
A pair of angles with a shared vertex and common
side but do not have overlapping interiors.
∠1 and ∠2 are adjacent. ∠3 and ∠4 are not.
∠1 and ∠ADC are not adjacent.
Adjacent Angles( a common side ) Non-Adjacent Angles
22°
36°
21
D
B
C
A
4
3
Definition:
Examples:

3. Lesson 1-5: Pairs of Angles 3
Complementary Angles
A pair of angles whose sum is 90˚Definition:
Examples:
Adjacent Angles
( a common side )
2
1
Q
A
B
C 1
2
Q
R
A
B
F
G
Non-Adjacent Angles
m∠1 = 40°m∠2 = 50°

4. Lesson 1-5: Pairs of Angles 4
Supplementary Angles
A pair of angles whose sum is 180˚Definition:
Examples:
Adjacent supplementary angles are
also called “Linear Pair.”
Non-Adjacent Angles
2 1
A Q
B
C
1
2
A Q
R
B
F
Gm∠2 = 140°
m∠1 = 40°

5. Lesson 1-5: Pairs of Angles 5
Vertical Angles
A pair of angles whose sides form opposite rays.Definition:
4
3
2
1
A
Q
D
B
C
Examples:
∠2 and ∠4
∠1 and ∠3
Vertical angles are non-adjacent angles formed by intersecting
lines.

6. Lesson 1-5: Pairs of Angles 6
Theorem: Vertical Angles are =
The diagramGiven:
4
3
2
1
A
Q
D
B
C
Prove:
~
∠1 ≅ ∠3
Statements Reasons
m∠2 + m∠3 = 180°
m∠1 + m∠2 = 180°1.
m∠1 + m∠2 = m∠2 + m∠32.
m∠1 = m∠33.
m∠1 ≅ m∠34.
1. Definition: Linear Pair
2. Property: Substitution
3. Property: Subtraction
4. Definition: Congruence

7. Lesson 1-5: Pairs of Angles 7
What’s “Important” in Geometry?
360˚ 180˚ 90˚
4 things to always look for !
. . . and Congruence
Most of the rules (theorems)
and vocabulary of Geometry
are based on these 4 things.

8. Lesson 1-5: Pairs of Angles 8
Example: If m∠4 = 67º, find the measures
of all other angles.
3 4 180m m∠ + ∠ = o
3 67 180m∠ + =o o
3 180 67 113m∠ = − = o
4
3
2
1
67º
Step 1: Mark the figure with given info.
Step 2: Write an equation.
3 1 , . 3 1 117∠ ∠ ∠ = ∠ = o
Because and are vertical angles they are equal m m
4 2 , . 4 2 67∠ ∠ ∠ = ∠ = o
Because and arevertical angles they are equal m m

9. Lesson 1-5: Pairs of Angles 9
Example: If m∠1 = 23 º and m∠2 = 32 º, find the
measures of all other angles.
4 23 ( 1 & 4 .)
5 32 ( 2 & 5 .)
m are vertical angles
m are vertical angles
∠ = ∠ ∠
∠ = ∠ ∠
o
o
6 5
4
32
1
Answers:
1 2 3 180
23 32 3 180
3 180 55 125
3 6 125
3 & 6 .
m m m
m
m
m m
are vertical angles
∠ + ∠ + ∠ =
+ + ∠ =
∠ = − =
∠ = ∠ =
∠ ∠
o
o o o
o
o

10. Lesson 1-5: Pairs of Angles 10
Example: If m∠1 = 44º, m∠7 = 65º find the
measures of all other angles.
3 90m∠ = o
1 4 44m m∠ = ∠ = o
4 5 90
44 5 90
5 46
m m
m
m
∠ + ∠ =
+ ∠ =
∠ =
o
o o
o 7
6
54
3
2 1
Answers:
6 7 90
6 65 90
6 25
m m
m
m
∠ + ∠ =
∠ + =
∠ =
o
o o
o

11. Lesson 1-5: Pairs of Angles 11
Algebra and Geometry
( ) = ( )
( ) + ( ) = ( )
( ) + ( ) = 90˚
( ) + ( ) = 180˚
Common Algebraic Equations used in Geometry:
If the problem you’re working on has a variable (x),
then consider using one of these equations.