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

 * Alternate Interior Angles.png

Let $AB$ and $CD$ be two infinite straight lines, and let $EF$ be a transversal that cuts them.

Let at least one pair of alternate interior angles, WLOG $\angle AHJ$ and $\angle HJD$, be equal.

Assume that the lines are not parallel.

Then they meet at some point $G$ which WLOG is on the same side as $B$ and $D$.

Since $\angle AHJ$ is an exterior angle of $\triangle GJH$, from External Angle of Triangle Greater than Internal Opposite, $\angle AHJ > \angle HJG$, a contradiction.

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

Therefore, by definition, they are parallel.