A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the opposite and interior angle, and the interior angles on the same side equal to two right angles.
Let EF fall on the parallel straight lines AB and CD.
Part 1: We will prove that the alternate angles AGH and GHD are equal.
If the two angles are not equal, then one must be greater. Let AGH be the greater angle.
Add angle BGH to both, and angles AGH and BGH are greater than angles GHD and BGH. But the angles AGH and BGH are equal to two right angles by Proposition 13 (If a straight line falls on a straight line, it makes two angles that are equal to two right angles.)
Therefore the two angles GHD and BGH are less than two right angles. Lines that produce indefinitely from angles less than two right angles meet (this is one of Euclid’s Postulates), meaning that AB and CD will meet.
But AB and CD cannot meet because they are parallel, and this contradiction implies that angle AGH cannot be greater than angle GHD.
Therefore the alternate angles AGH and GHD must be equal.
Part 2: We will prove that the exterior angle EGB is equal to the interior and opposite angle GHD.
In Part 1 we proved that angle AGH is equal to angle GHD.
By Proposition 15 (vertical angles are equal to each other) angle AGH is equal to angle EGB.
Therefore the exterior angle EGB is equal to the interior and opposite angle GHD.
Part 3: We will prove that the interior angles on the same side (BGH and GHD) are equal to two right angles.
In Part 2 we proved that angle EGB was equal to angle GHD. Add angle BGH to each, and we have that the two angles EGB and BGH are equal to the two angles GHD and BGH.
Again using Proposition 13 (a straight line that falls on a straight line produces two angles that are equal to two right angles), the two angles EGB and BGH are equal to two right angles. That implies that the two angles BGH and GHD are equal to two right angles.
Therefore the interior angles on the same side are equal to two right angles.
Q.E.D.
More fun with parallel lines in the next Proposition – George
