1. Identify the type of angles.
, 7
ANSWER
corresponding
, 6
ANSWER
alternate interior
, 8
ANSWER
alternate exterior

2. Use Properties of Parallel and Perpendicular Lines
Target
Use Properties of Parallel and Perpendicular Lines
You will…
Use angles formed by parallel lines and transversals.

3. VOCABULARY
Corresponding Angles Postulate 15 – If two parallel lines are cut by a transversal, then pairs of corresponding angles are congruent. 1
Proof sequence: First, identify the angles as corresponding by the definition of corresponding angles; then state the angles are congruent by the Corresponding Angles Postulate 15.

4. VOCABULARY
Alternate Interior Angles Theorem 3.1 – If two parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent.
Alternate Exterior Angles Theorem 3.2 – If two parallel lines are cut by a transversal, then pairs of alternate exterior angles are congruent.
Consecutive Interior Angles Theorem 3.3 – If two parallel lines are cut by a transversal, then pairs of consecutive interior angles are supplementary.
Proof sequence: First, identify the angle pair by type, then state the angles are congruent or supplementary by one of the theorems about parallel lines.

5. EXAMPLE 1
Identify congruent angles
The measure of three of the numbered angles is 120°. Identify the angles. Explain your reasoning.
SOLUTION
By the Corresponding Angles Postulate, m 5 = 120°. Using the Vertical Angles Congruence Theorem, m 4 = 120°. Because and are corresponding angles, by the Corresponding Angles Postulate, you know that m = 120°.

6. Use properties of parallel lines
EXAMPLE 2
Use properties of parallel lines
ALGEBRA
Find the value of x.
SOLUTION
By the Vertical Angles Congruence Theorem, m = 115°. Lines a and b are parallel, so you can use the theorems about parallel lines.
m (x+5)° =
180°
Consecutive Interior Angles Theorem
115° + (x+5)° =
180°
Substitute 115° for m
x =
180
Combine like terms.
x = 60
Subtract 120 from each side.

7. GUIDED PRACTICE for Examples 1 and 2 Use the diagram. 1.
If m = 105°, find m 4, m 5, and m Tell which postulate or theorem you use in each case.
m =
105°
ANSWER
Vertical Angles Congruence Theorem.
m =
105°
Corresponding Angles Postulate.
m =
105°
Alternate Exterior Angles Theorem

8. GUIDED PRACTICE
for Examples 1 and 2
Use the diagram. 2.
If m = 68° and m = (2x + 4)°, what is the value of x? Show your steps.
m m =
180
ANSWER
m =
m 7
68 + 2x + 4 =
180
2x + 72 =
180
2x =
108
x = 54

9. EXAMPLE 3
Prove the Alternate Interior Angles Theorem
Prove that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
SOLUTION
Draw a diagram. Label a pair of alternate interior angles as 1 and You are looking for an angle that is related to both 1 and Notice that one angle is a vertical angle with and a corresponding angle with Label it

10. Prove the Alternate Interior Angles Theorem
EXAMPLE 3
Prove the Alternate Interior Angles Theorem
GIVEN :
p q
PROVE :
∠ ∠ 2
STATEMENTS
REASONS
p q 1. 1.
Given 2.
Corresponding Angles Postulate 2.
1 ∠ 3.
3 ∠ 3.
Vertical Angles Congruence Theorem 4.
1 ∠
Transitive Property of Congruence 4.

11. EXAMPLE 4
Solve a real-world problem
Science
When sunlight enters a drop of rain, different colors of light leave the drop at different angles. This process is what makes a rainbow. For violet light, m = 40°. What is m 1? How do you know?

12. EXAMPLE 4
Solve a real-world problem
SOLUTION
Because the sun’s rays are parallel, 1 and 2 are alternate interior angles. By the Alternate Interior Angles Theorem, By the definition of congruent angles, m = m = 40°.