Equal Alternate Interior Angles Implies Parallel

 

 

Jump to: navigation, search

It has been suggested that this article or section be renamed: Implies → implies; perhaps add "Lines" at the end? One may discuss this suggestion on the talk page.

Theorem

Given two infinite straight lines which are cut by a transversal, if the alternate interior angles are equal, then the lines are parallel.

Proof

Let AB and CD be two straight lines, and let EF be a transversal that cuts them. Let the at least one pair of alternate interior angles, WLOG ∠AEF and ∠EFD , be equal.
Assume that the lines are not parallel. Then the meet at some point G which WLOG is on the same side as B and D .
Since ∠AEF is an exterior angle of △GEF , from External Angle of Triangle Greater than Internal Opposite, ∠AEF>∠EFG , a contradiction.
Similarly, they cannot meet on the side of A and C .
Therefore, by definition, they are parallel.

 ProofWiki