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Lesson 3.2
PROOFS FOR PARALLEL LINES
Assuming that you have your own notes on Chapter three, I only need to point out a few specific ideas. A
transversal is a line that intersects two coplanar lines at two distinct points. When those lines are parallel, the
angles have certain properties that we use often in geometry. The three types of angles that we will look at are
Corresponding Angles, Alternate Interior Angles, Consecutive Interior Angles, and Alternate Exterior Angles.
Remember, a postulate is something that we can
assume to be true and we do not have to prove it, thus,
the Corresponding angles postulate is a nice way to
start this section.
Look at the alternate interior angles theorem on page 148. Look at the two column proof below and see if you
understand the reasoning and can supply the missing parts if the numbers and diagram differ from the textbook.
STATEMENTS
1)
2)
3)
4)
REASONS
1) GIVEN
2)
3)
4) TRANSITIVE PROPERTY OF CONGRUENCE
Using alternate interior angles and corresponding angles formed by parallel lines.
YOU TRY TO PROVE THE CONSECUTIVE INTERIOR ANGLES THEOREM.
This differs from the diagram, so follow and complete this proof.
Lesson 3.2
Given: a ll b
Prove: <1 and <2 are supplementary
Statements
1) a ll b
2) <2 and <3 are a linear pair
3) <2 and <3 are supplementary
4)m< + m< =180
5) 1  3
6) m<1=m<3
7) m<2 + m<1 = 180
8) <1 and <2 are supplementary
Prove the alternate exterior angle theorem.
Putting it all together
Reasons
1) Given
2) Def of
3)
4) Definition of supplementary angles
5) Corresponding Angles Postulate
6) Def of
7)
8)
Lesson 3.2