Parallelism implies Supplementary Interior Angles

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

Proof

 * Parallel Cut by Transversal.png

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

Let $EF$ be a transversal that cuts them.

From Parallelism implies Equal Corresponding Angles and Euclid's second common notion:
 * $\angle EGB + \angle BGH = \angle DHG + \angle BGH$

From Two Angles on Straight Line make Two Right Angles, $\angle EGB + \angle BGH$ equals two right angles.

So by definition, $\angle BGH$ and $\angle DHG$ are supplementary.