Equal Corresponding Angles or Supplementary Interior Angles implies Parallel Lines

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Theorem

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 $28$)


Equal Corresponding Angles implies Parallel Lines

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


Supplementary Interior Angles implies Parallel Lines

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.


Historical Note

This theorem is Proposition $28$ of Book $\text{I}$ of Euclid's The Elements.
This theorem (which has two parts) is the converse of the second and third parts of Proposition $29$: Parallelism implies Equal Alternate Interior Angles, Corresponding Angles, and Supplementary Interior Angles.


Sources