Parallelism implies Supplementary Interior Angles

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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.


In the words of Euclid:

A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.

(The Elements: Book $\text{I}$: Proposition $29$)


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.

$\blacksquare$


Historical Note

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


Sources