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.

In the words of Euclid:

If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the the same side equal to two right angles, the straight lines will be parallel to one another.

(The Elements: Book $\text{I}$: Proposition $28$)

Proof

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.

Without loss of generality, 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 Angles implies Parallel Lines:

$AB \parallel CD$

$\blacksquare$

Historical Note

This proof is the second part of Proposition $28$ of Book $\text{I}$ of Euclid's The Elements.
It is the converse of the third part of Proposition $29$: Parallelism implies Supplementary Interior Angles.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions