Proposition 14
If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another.
[Desmos graphs can be found here]
We will start with a straight line AB, and from point B we will draw two lines BC and BD not lying on the same side. The adjacent angles ABC and ABD are equal to two right angles. We will show that BC and BD lie on a straight line.
Assume that BD is not on a straight line with BC. Let BE lie in a straight line with BC.
By Proposition 13 (If a straight line is set up on a straight line, it will make two right angles or two angles equal to two right angles), since AB lies on the straight line CBE the two angles ABC and ABE are equal to two right angles.
We began with ABC and ABD being equal to two right angles, so the two angles ABC and ABE are equal to the two angles ABC and ABD.
Subtract angle ABC from both, leaving angle ABE equal to angle ABD which is a contradiction. Therefore BE does not lie on a straight line with BC. There cannot be any other straight line except for BD that lies on a straight line with BC.
Therefore BC lies on a straight line as BD.
Q.E.D.