Two Angles of Triangle are Less than Two Right Angles

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

Proof


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 $$\triangle ABC$$.

Note
This is Proposition 17 of Book I of Euclid's "The Elements".