Two Angles on Straight Line make Two Right Angles

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

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 proof 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

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions
• 1968: M.N. Aref and William Wernick: Problems & Solutions in Euclidean Geometry ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.1$