Euclid's Elements - Book 1 - Proposition 17
Any two angles of a triangle are together less than two right angles.
In other words: Given any triangle ABC, the sum of any of
the two inner angles is less than two right angles.
1. Given triangle ABC with angles , , and .
2. Extend line BD to point D, with external angle .
3. The sum of the angles ACB and ACD are equal to two right angles (I.13).
4. The angle ACD is greater than either angle ABC or CAB (I.16).
5. Add the same value (angle ACB) to both sides of each inequality.
6. Therefore, the sum of either ABC or CAB with angle ACB will be less than 180 (or 2 right angles).
Q.E.D.