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Chapter 3 Lesson Chapter 3 Lesson 22
Objective:Objective: To use a transversal in proving lines
parallel.
Postulate 3-2:Postulate 3-2:
Converse of the Corresponding Angles Postulate
If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel.
Theorem 3-3:Theorem 3-3:
Converse of the Alternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel.
l
m
11
22l ml m
l
m11
2244
If 1 2, then l m.
Theorem 3-4:Theorem 3-4:
Converse of the Same-Side Interior Angles Theorem
If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. If 2 and 4 are
supplementary, then l m
l
m11
2244
Example 1:Example 1:Proving Theorem 3-3Proving Theorem 3-3
Statements Reasons
1.) 1.)
2.) 2.)
3.) 3.)
4.) 4.)
Given:Given: 1 2 1 2 Prove:Prove: l ml m
l
m11
22
33
1 21 2 1 31 3 3 23 2
l ml m
Given
Vertical angles are congruentTransitive Prop. Of Congruence
Postulate 3-2
Example 2:Example 2:Using Theorem 3-4Using Theorem 3-4
Which lines, if any, must be parallel if 1 2? Justify Which lines, if any, must be parallel if 1 2? Justify your answer with a theorem or postulate.your answer with a theorem or postulate.
EE
DD
CC
KK11
33
44
22
DE is parallel to KC by DE is parallel to KC by Theorem 3-3, the Theorem 3-3, the
Converse of Alternate Converse of Alternate Interior Angles Interior Angles
Theorem.Theorem.
Example 3:Example 3:Which lines, if any, must be parallel if 3 4? Justify Which lines, if any, must be parallel if 3 4? Justify your answer with a theorem or postulate.your answer with a theorem or postulate.
EE
DD
CC
KK11
33
44
22
EC is parallel to DK by EC is parallel to DK by Postulate 3-2, Converse Postulate 3-2, Converse
of Corresponding of Corresponding Angles Postulate.Angles Postulate.
Example 4:Example 4:Proof of Theorem 3-6Proof of Theorem 3-6
Statements Reasons
1.) 1.)
2.) 2.)
3.) 3.)
4.)
5.) r sr s
4.)
5.)
Given:Given: tstr ,
Prove:Prove: rr ss
s
r
Given
Given
All Right Angles are Congruent
Converse of Corresponding Angles Postulate
t
1
2
tr ts
Angles 1 & 2 are right angles Definition of Perpendicular lines
21
l
m (2x+6)(2x+6)
4040
Example 5:Example 5:Find the value of x for which l m.
Corresponding Angles
2x+6 = 40
2x=34
x=17
Assignment:Assignment:
pg.125-127 #1-35
DUE Monday
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