Parallelism implies Equal Alternate Angles

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

Proof

 * Parallel Cut by Transversal.png

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

Let $EF$ be a transversal that cuts them.

Aiming for a contradiction, suppose the alternate angles are not equal.

Then one of the pair $\angle AGH$ and $\angle GHD$ must be greater.

, let $\angle AGH$ be greater.

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

So $\angle GHD + \angle BGH$ is less than two right angles.

By Euclid's fifth postulate, lines extended infinitely from angles less than two right angles must meet.

But the lines are parallel.

So by definition the lines do not intersect.

From this contradiction it follows that the alternate angles are be equal.