Equal Alternate Angles implies Parallel Lines

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 $$\angle AEF$$ and $$\angle EFD$$, 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 $$\angle AEF$$ is an exterior angle of $$\triangle GEF$$, from External Angle of Triangle Greater than Internal Opposite, $$\angle AEF > \angle EFG$$, a contradiction.

Similarly, they cannot meet on the side of $$A$$ and $$C$$.

Therefore, by definition, they are parallel.