Two Angles on Straight Line make Two Right Angles

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Theorem

In the words of Euclid:

If a straight line set up on a straight line make angles, it will make either two right angles or two angles equal to two right angles.

(The Elements: Book $\text{I}$: Proposition $13$)


Proof

Euclid-I-13.png

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


If $\angle ABC = \angle ABD$ then from Book $\text{I}$ Definition $10$: Right Angle they are two right angles.

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

Then:

$\angle CBE = \angle EBD$

which from Book $\text{I}$ Definition $10$: Right Angle 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.

$\blacksquare$


Historical Note

This theorem is Proposition $13$ of Book $\text{I}$ of Euclid's The Elements.
It is the converse of Proposition $14$: Two Angles making Two Right Angles make Straight Line.


Sources