Two Angles on Straight Line make Two Right Angles

Theorem
If a straight line set up on another straight line so as to make angles, it makes either two right angles or two angles which add up to two right angles.

Proof


Let the line $$AB$$ be set up on the line $$CD$$.

If $$\angle ABC = \angle ABD$$ then from Definition I-10 they are two right angles.

If not, we draw $BE$ at right angles to $CD$.

Then $$\angle CBE = \angle EBD$$ which from Definition I-10 are both right angles.

Since $$\angle CBE = \angle ABC + \angle ABE$$, we add $$\angle EBD$$ to each.

Therefore by Common Notion 2 $$\angle CBE + \angle EBD = \angle ABC + \angle ABE + \angle EBD$$.

Therefore $$\angle ABC + \angle ABD = \angle ABC + \angle ABE + \angle EBD$$.

Again, since $$\angle ABD = \angle EBD + \angle ABE$$, we add $$\angle ABC$$ to each.

Therefore by Common Notion 2 $$\angle ABD + \angle ABC = \angle ABC + \angle ABE + \angle EBD$$.

But $$\angle CBE + \angle EBD$$ equals the same three angles.

Therefore by Common Notion 1 $$\angle CBE + \angle EBD = \angle ABC + \angle ABD$$.

But $$\angle CBE$$ and $$\angle EBD$$ are both right angles.

Hence the result.

Note
This is Proposition 13 of Book I of Euclid's "The Elements".

This theorem is the converse of Proposition 14: Two Angles making Two Right Angles make a Straight Line.