Supplementary Interior Angles implies Parallel Lines

Theorem
Given two infinite straight lines which are cut by a transversal, if the interior angles on the same side of the transversal are supplementary, 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 interior angles on the same side of the transversal be supplementary.

, let those interior angles be $\angle BGH$ and $\angle DHG$.

So, by definition, $\angle DHG + \angle BGH$ equals two right angles.

Also, from Two Angles on Straight Line make Two Right Angles, $\angle AGH + \angle BGH$ equals two right angles.

Then from Euclid's first and third common notion and Euclid's fourth postulate:
 * $\angle AGH = \angle DHG$

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