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

 * Euclid-I-13.png

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.