Equal Corresponding Angles implies Parallel Lines

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Theorem

Given two infinite straight lines which are cut by a transversal, if the corresponding angles are equal, 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

Parallel Cut by Transversal.png

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

Let $EF$ be a transversal that cuts them.

Let at least one pair of corresponding angles be equal.

Without loss of generality, let $\angle EGB = \angle GHD$.

By the Vertical Angle Theorem:

$\angle GHD = \angle EGB = \angle AGH$

Thus by Equal Alternate Interior Angles implies Parallel Lines:

$AB \parallel CD$

$\blacksquare$


Historical Note

This theorem is the first part of Proposition $28$ of Book $\text{I}$ of Euclid's The Elements.
It is the converse of the second part of Proposition $29$: Parallelism implies Equal Corresponding Angles.


Sources