Equal Corresponding Angles implies Parallel Lines

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

Proof

 * Parallel Cut by Transversal.png

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

Let $EF$ be a transversal that cuts them.

Let at least one pair of corresponding angles, WLOG $\angle EGB$ and $\angle GHD$, be equal.

By the Vertical Angle Theorem:
 * $\angle GHD = \angle EGB = \angle AGH$

Thus by Equal Alternate Interior Angles implies Parallel Lines:
 * $AB \parallel CD$