Two Angles of Triangle Less than Two Right Angles

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Theorem

In the words of Euclid:

In any triangle two angles taken together in any manner are less than two right angles.

(The Elements: Book $\text{I}$: Proposition $17$)


Proof

Euclid-I-17.png

Let $\triangle ABC$ be a triangle.

Let the side $BC$ be extended to $D$.

Since the angle $\angle ACD$ is an external angle of $\triangle ABC$, it follows that it is greater than both $\angle BAC$ and $\angle ABC$.

We add $\angle ACB$ to both, so that $\angle ACD + \angle ACB$ is greater than $\angle ABC + \angle ACB$.

But $\angle ACD + \angle ACB$ is equal to two right angles.

Therefore $\angle ABC + \angle ACB$ is less than two right angles.


In a similar manner we show that the same applies to the other two pairs of internal angles of $\triangle ABC$.

$\blacksquare$


Historical Note

This theorem is Proposition $17$ of Book $\text{I}$ of Euclid's The Elements.


Sources