Lesson 1-5: Pairs of Angles 1

Lesson 1-5

Pairs of Angles

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

A4

3

Definition:

Examples:

Lesson 1-5: Pairs of Angles 3

Complementary AnglesA pair of angles whose sum is 90˚Definition:

Examples:

Adjacent Angles( a common side )

21

Q

AB

C 1

2

Q

R

AB

F

G

Non-Adjacent Angles

m1 = 40°m2 = 50°

Lesson 1-5: Pairs of Angles 4

Supplementary AnglesA 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 QR

BF

Gm2 = 140°

m1 = 40°

Lesson 1-5: Pairs of Angles 5

Vertical AnglesA pair of angles whose sides form opposite rays.Definition:

4

3

2

1A

Q

D

B

C

Examples:

2 and 4

1 and 3

Vertical angles are non-adjacent angles formed by intersecting lines.

Lesson 1-5: Pairs of Angles 6

Theorem: Vertical Angles are =

The diagramGiven:

4

3

2

1A

Q

D

B

C

Prove:

~

1 3

Statements Reasons

m2 + m3 = 180°m1 + m2 = 180°1.

m1 + m2 = m2 + m32.

m1 = m33.

m1 m34.

1. Definition: Linear Pair

2. Property: Substitution

3. Property: Subtraction

4. Definition: Congruence

Lesson 1-5: Pairs of Angles 7

What’s “Important” in Geometry?

360˚ 180˚ 90˚

4 things to always look for !

. . . and CongruenceMost of the rules (theorems)and vocabulary of Geometryare based on these 4 things.

Lesson 1-5: Pairs of Angles 8

Example: If m4 = 67º, find the measures of all other angles.

3 4 180m m

3 67 180m

3 180 67 113m

4

3

2

1

67º

Step 1: Mark the figure with given info.

Step 2: Write an equation.

3 1 , . 3 1 117 Because and are vertical angles they are equal m m

4 2 , . 4 2 67 Because and arevertical angles they are equal m m

Lesson 1-5: Pairs of Angles 9

Example: If m1 = 23 º and m2 = 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

6 5

4 3 2

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

Lesson 1-5: Pairs of Angles 10

Example: If m1 = 44º, m7 = 65º find the measures of all other angles.

3 90m

1 4 44m m

4 5 90

44 5 90

5 46

m m

m

m

7

6 5 4

3

2 1

Answers:

6 7 90

6 65 90

6 25

m m

m

m

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.