Two Angles of Triangle Less than Two Right Angles

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 $\triangle ABC$.