Proofs

These results can be used to prove two lines are parallel or to prove results about angles formed by a transversal cutting parallel lines.

Example 1:

Given:	,
Prove:

Proof:

 and  are alternate interior angles for lines AC and DE with transversal CE, so sides AC and DE are parallel.  Likewise,  and  are alternate interior angles for lines AE and CD with transversal AD, so sides AE and CD are parallel.  Therefore ABCD is a parallelogram because its opposite sides are parallel.

Example 2:

Given:	Angles P and Q are supplementary.
Prove:	Side PS is parallel to side QR.

Proof: 

Angles P and Q are same-side interior angles for lines PS and QR with transversal PQ.  Since these angles are supplementary, sides PS and QR must be parallel.

Example 3:

Given:	is supplementary to , line is parallel to line
Prove:

Proof:

 and  are same-side interior angles for lines  and .  Since these lines are parallel,  and  are supplementary.  Since  is also supplementary to  and  is congruent to itself (reflexive property),  (supplements of congruent angles are congruent).

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