1. 3.2 Use Parallel Lines and Transversals

2. Objectives Use the properties of parallel lines to determine
congruent angles
Use algebra to find
angle measures

3. Postulate 15 Corresponding s Postulate
If 2  lines are cut by a transversal, then each pair of corres. s is .
i.e. If l m, then 12. 1 2 l m

4. Theorem 3.1 Alternate Interior s Theorem
If 2  lines are cut by a transversal, then each pair of alternate interior s is .
i.e. If l m, then 12. 1 2 l m

5. Theorem 3.2 Alternate Exterior s Theorem
If 2  lines are cut by a transversal, then the pairs of alternate exterior s are .
i.e. If l m, then 12.
l m 1 2

6. Theorem 3.3 Consecutive Interior s Theorem
If 2  lines are cut by a transversal, then each pair of consecutive int. s is supplementary.
i.e. If l m, then 1 & 2 are supplementary or m1 + m2 = 180°. l m 1 2

7. Theorem 3.11  Transversal Theorem
If a transversal is  to one of 2  lines, then it is  to the other.
i.e. If l m, & t  l, then t m. t 1 2 l m

8. EXAMPLE 1
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°.

9. Use Properties of Parallel Lines
EXAMPLE 2
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.

10. YOUR TURN GUIDED PRACTICE 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

11. YOUR TURN
GUIDED PRACTICE
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

12. Prove the Alternate Interior Angles Theorem
EXAMPLE 3
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
GIVEN :
p q
PROVE :
∠ ∠ 2