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.