Parallelism implies Equal Alternate 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 alternate interior angles are equal.


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.

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

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

Without loss of generality, 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 interior angles are be equal.

$\blacksquare$


Historical Note

This theorem is the first part of Proposition $29$ of Book $\text{I}$ of Euclid's The Elements.
It is the converse of Proposition $27$: Equal Alternate Interior Angles implies Parallel Lines.

This is the first proposition of The Elements to make use of Euclid's fifth postulate.


Sources