Parallelism implies Equal Corresponding Angles

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Theorem

Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the corresponding 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.

From Parallelism implies Equal Alternate Interior Angles:

$\angle AGH = \angle DHG$

By the Vertical Angle Theorem:

$\angle EGB = \angle AGH = \angle DHG$

$\blacksquare$


Historical Note

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


Sources