Equal Alternate Angles implies Parallel Lines

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

Proof

 * Transversal.png

Let $AB$ and $CD$ be two infinite straight lines.

Let $EF$ be a transversal that cuts them.

Let at least one pair of alternate angles be equal.

, let $\angle AHJ = \angle HJD$.

that $AB$ and $CD$ are not parallel.

Then they meet at some point $G$.

, let $G$ be on the same side as $B$ and $D$.

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

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

Therefore, by definition, $AB$ and $CD$ are parallel.