Equal Alternate Interior Angles Implies Parallel

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.

$\blacksquare$

Historical Note

This is Proposition 27 of Book I of Euclid's The Elements.

This theorem is the converse of the first part of Proposition 29.