Two Angles on Straight Line make 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 they are two right angles.

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

Then:
 * $\angle CBE = \angle EBD$

which from are both right angles.

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

Therefore by :
 * $\angle CBE + \angle EBD = \angle ABC + \angle ABE + \angle EBD$

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 :
 * $\angle ABD + \angle ABC = \angle ABC + \angle ABE + \angle EBD$

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

Therefore by :
 * $\angle CBE + \angle EBD = \angle ABC + \angle ABD$

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

Hence the result.