Triangle Angle Theorems

Every triangle's angles add to 180°. That is worth seeing proved rather than just accepted: draw a line through one vertex parallel to the opposite side, and the alternate interior angles move the other two angles up to form a straight line.

The Exterior Angle Theorem is the time-saver. Extend one side of a triangle and the angle formed outside equals the sum of the two angles it is *not* touching.

If the two far angles are 50° and 60°, the exterior angle is 110° — no subtraction from 180 needed.

Triangles are classified by angles (acute, right, obtuse) and by sides (scalene, isosceles, equilateral). A triangle can have at most one right or obtuse angle, since two would already reach or exceed 180°.

The Triangle Sum Theorem, m∠A + m∠B + m∠C = 180°, is a consequence of the Parallel Postulate: the auxiliary line through one vertex parallel to the opposite side creates two pairs of congruent alternate interior angles that together with the third angle form a straight angle.

The Exterior Angle Theorem follows immediately: an exterior angle is supplementary to its adjacent interior angle, and the three interior angles sum to 180°, so the exterior angle equals the sum of the two remote interior angles. A corollary is that an exterior angle is strictly greater than either remote interior angle, which is the inequality used in indirect proofs.

Two further corollaries: the acute angles of a right triangle are complementary, and each angle of an equiangular triangle is 60°.

Generalising to convex polygons, the interior sum is (n − 2)·180° while the exterior sum is 360° for every n, since traversing the boundary turns through one full revolution.