Equal Corresponding Angles or Supplementary Interior Angles Implies Parallel

Theorem

Part 1

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

Part 2

Given two infinite straight lines which are cut by a transversal, if the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

Proof

Part 1

Let $AB$ and $CD$ be infinite straight lines, and let $EF$ be a transversal that cuts them. Let at least one pair of corresponding angles, WLOG $\angle EGB$ and $\angle GHD$, be equal.

$\angle GHD = \angle EGB = \angle AGH$ by the Vertical Angle Theorem.

Thus $AB \parallel CD$ by Equal Alternate Interior Angles Implies Parallel.

$\blacksquare$

Part 2

Let $AB$ and $CD$ be infinite straight lines, and let $EF$ be a transversal that cuts them. Let at least one pair of interior angles on the same side of the transversal, WLOG $\angle BGH$ and $\angle DHG$ be supplementary, so by definition $\angle DHG + \angle BGH$ equals two right angles.

$\angle AGH + \angle BGH$ equals two right angles.

Then from Euclid's first and third common notion and Euclid's fourth postulate, $\angle AGH = \angle DHG$.

Finally, $AB \parallel CD$ by Equal Alternate Interior Angles Implies Parallel.

$\blacksquare$

Historical Note

This is Proposition 28 of Book I of Euclid's The Elements.

This theorem is the converse of the second and third parts of Proposition 29.