Theorem. Any two angles (B, C) of a triangle (ABC) are together less than two right angles.
Dem.—Produce BC to D; then the exterior angle ACD is greater than ABC [xvi.]: to each add the angle ACB, and we have the sum of the angles ACD, ACB greater than the sum of the angles ABC, ACB; but the sum of the angles ACD, ACB is two right angles [xiii.]. Therefore the sum of the angles ABC, ACB is less than two right angles.
In like manner we may show that the sum of the angles A, B, or of the angles A, C, is less than two right angles.
Cor. 1.—Every triangle must have at least two acute angles.
Cor. 2.—If two angles of a triangle be unequal, the lesser must be acute.
Exercise.
Prove Prop. xvii. without producing a side.