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 same side equal to two right angles, the straight lines will be parallel to one another.
Let the straight line EF fall on the two straight lines AB and CD, intersecting AB at a point G and intersecting CD at a point H, is such a way that exterior angle EGB is equal to the interior and opposite angle GHD or the interior angles on the same side (BGH and GHD) are equal to two right angles.
Starting with angle EGB being equal to angle GHD we will prove that AB is parallel to CD.
- Angle EGB is equal to angle GHD by hypothesis.
- Angle EGB is also equal to angle AGH by Proposition 15 (vertical angles are equal)
So, angle GHD is equal to angle AGH. Since these two angles are alternate angles, AB and CD are parallel by Proposition 27. (If a line falling on two straight lines produces equal alternate angles then the two lines are parallel.)
Now we will start with angles BGH and GHD being equal to two right angles, and will once again prove that AB and CD are parallel.
Angles AGH and BGH are also equal to two right angles by Proposition 13. (A straight line set up on a straight line will make angles equal to two right angles.)
So, the angles AGH and BGH together are equal to the angles BGH and GHD together. Subtracting angle BGH from each, the remaining angle AGH is equal to the remaining angle GHD.
Since angles AGH and GHD are alternate angles, Proposition 27 again tells us that AB and CD are parallel.
Q.E.D.