Two Angles making Two Right Angles make a Straight Line

Theorem

Given a straight line and a point on it, let two straight lines, not lying on the same side of it, make the adjacent angles equal to two right angles, those two straight lines will be in a straight line with one another.

Proof

Let $AB$ be the given straight line and $B$ the given point on it.

Let $BC$ and $BD$ be the two straight lines not lying on the same side of it such that $\angle ABC + \angle ABD$ equal two right angles.

Suppose $BD$ is not in a straight line with $BC$.

Now some line must be, so let $BE$ be in a straight line with $BC$ instead.

Then since $AB$ stands on the (supposed) straight line $CBE$, $\angle ABC + \angle ABE$ equal two right angles.

But we already know that $\angle ABC + \angle ABD$ equal two right angles.

Therefore, by Common Notion 1 and the fact that all right angles are congruent, $\angle ABC + \angle ABD = \angle ABC + \angle ABE$.

So let $\angle ABC$ be subtracted from each.

Then by Common Notion 3, the remaining angle $\angle ABE$ equals remaining angle $\angle ABD$.

But $\angle ABD$ is greater than $\angle ABE$, so this is impossible.

So $BE$ can not be in a straight line with $BC$.

Similarly we can show that any straight line which is not $BD$ can not be in a straight line with $BC$.

Therefore $BC$ is in a straight line with $BD$, hence the result.

$\blacksquare$

Historical Note

This is Proposition 14 of Book I of Euclid's The Elements.

This theorem is the converse of Proposition 13: Two Angles on a Straight Line make Two Right Angles.