Equal Alternate Interior Angles implies Parallel Lines

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Theorem

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


In the words of Euclid:

If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.

(The Elements: Book $\text{I}$: Proposition $27$)


Proof

Alternate Interior Angles.png

Let $AB$ and $CD$ be two infinite straight lines, and let $EF$ be a transversal that cuts them.

Let at least one pair of alternate interior angles be equal.

Without loss of generality, let $\angle AHJ = \angle HJD$.

Assume that the lines are not parallel.

Then they meet at some point $G$.

Without loss of generality, let $G$ be on the same side as $B$ and $D$.

Since $\angle AHJ$ is an exterior angle of $\triangle GJH$, from External Angle of Triangle Greater than Internal Opposite, $\angle AHJ > \angle HJG$, a contradiction.

Similarly, they cannot meet on the side of $A$ and $C$.

Therefore, by definition, they are parallel.

$\blacksquare$


Historical Note

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


Sources