Parallel Implies Equal Alternate Interior Angles, Corresponding Angles, and Supplementary Interior Angles

Theorem

As Euclid defined it:

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 I: Proposition $29$)

Worded in more contemporary language:

Part 1

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

Part 2

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

Part 3

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.

Proof

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

Part 1

Assume the alternate interior angles are not equal. Then one of the pair $\angle AGH$ and $\angle GHD$ must be greater. WLOG let $\angle AGH$ be greater.

$\angle AGH + \angle BGH$ equal two right angles, so $\angle GHD + \angle BGH$ is less than two right angles.

Lines extended infinitely from angles less than two right angles must meet by Euclid's fifth postulate.

But the lines are parallel, so by definition the lines do not intersect, a contradiction.

Thus, the alternate interior angles must be equal.

$\blacksquare$

Part 2

From part 1, $\angle AGH = \angle DHG$.

By the Vertical Angle Theorem, $\angle EGB = \angle AGH = \angle DHG$.

$\blacksquare$

Part 3

From part 2 and Euclid's second common notion, $\angle EGB + \angle BGH = \angle DHG + \angle BGH$.

$\angle EGB + \angle BGH$ equal two right angles, so by definition $\angle BGH$ and $\angle DHG$ are supplementary.

$\blacksquare$

Historical Note

This is Proposition 29 of Book I of Euclid's The Elements.

The first part of this theorem is the converse of Proposition 27 and the second and third parts are the converse of Proposition 28.

Also note that this is the first proposition of The Elements to make use of Euclid's fifth postulate.