3.2 Use Parallel Lines and
Transversals
OBJECTIVES
 Use
the properties of
parallel lines to determine
congruent angles
 Use
algebra to find
angle measures
POSTULATE 15
CORRESPONDING S
POSTULATE

If 2  lines are cut by a
transversal, then each
pair of corres. s is .
1
2
l

i.e. If l m, then 12.
m
THEOREM 3.1
ALTERNATE INTERIOR S
THEOREM

If 2  lines are cut by a transversal, then each pair of
alternate interior s is .
l
1
2
m

i.e. If l m, then 12.
THEOREM 3.2
ALTERNATE EXTERIOR S
THEOREM

If 2  lines are cut by a transversal, then the pairs of
alternate exterior s are .
l

m
1
2

i.e. If l m, then 12.
THEOREM 3.3
CONSECUTIVE INTERIOR S
THEOREM

If 2  lines are cut by a transversal, then each pair of
consecutive int. s is supplementary.
l
1
m
2
i.e. If l m, then 1 & 2 are supplementary or m1 + m2 = 180°.
THEOREM 3.11
 TRANSVERSAL THEOREM

If a transversal is  to one of 2  lines, then it is  to
the other.
t
l
m

1
2
i.e. If l m, & t  l, then t m.
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 4 and 8 are corresponding
angles, by the Corresponding Angles Postulate, you
know that m 8 = 120°.
EXAMPLE 2
EXAMPLE 2
ALGEBRA
Use Properties of Parallel Lines
Find the value of x.
SOLUTION
By the Vertical Angles Congruence Theorem,
m 4 = 115°. Lines a and b are parallel, so you can
use the theorems about parallel lines.
m
4 + (x+5)° = 180°
115° + (x+5)° = 180°
x + 120 = 180
x = 60
Consecutive Interior Angles Theorem
Substitute 115° for m
4.
Combine like terms.
Subtract 120 from each side.
YOUR TURN
GUIDED PRACTICE
Use the diagram.
1. If m 1 = 105°, find m 4, m 5, and m 8. Tell
which postulate or theorem you use in each
case.
ANSWER
m
4 = 105°
Vertical Angles Congruence Theorem.
m
5 =105°
Corresponding Angles Postulate.
m
8 =105°
Alternate Exterior Angles Theorem
YOUR TURN
GUIDED PRACTICE
Use the diagram.
2. If m 3 = 68° and m 8 = (2x + 4)°, what is the
value of x? Show your steps.
ANSWER
m
7+m
m
8 = 180
3= m
68 + 2x + 4 = 180
2x + 72 = 180
2x = 108
x = 54
7
EXAMPLE 3
EXAMPLEProve
3 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 2. You are looking for an angle
that is related to both 1 and 2. Notice that one
angle is a vertical angle with 2 and a corresponding
angle with 1. Label it 3.
GIVEN :
p q
PROVE :  1  2