Proof of the Angle Sum Theorem

It doesn't matter whether the triangle is small or large, acute, right or obtuse. The total is always 180°.

This makes it easy to find a missing angle. If two angles of a triangle measure 50° and 60°, the third angle is 180° − 50° − 60° = 70°.

On this page you will first see why the theorem works with a numerical example, and then prove it for every triangle.

Three facts the proof uses

1. Parallel line through a point. Through a point that is not on a line, you can draw exactly one line parallel to that line.
2. Alternate interior angles. When a line (called a transversal) crosses two parallel lines, the alternate interior angles are equal. We accept this fact without proof.
3. Straight angle. Angles that together form a straight line add up to 180°.

Alternate interior angles

In the figure below, two parallel lines are cut by a transversal. Alternate interior angles lie between the parallel lines, on opposite sides of the transversal.

The figure shows two pairs of alternate interior angles. For the red pair, ∠1 = ∠2. For the blue pair, ∠3 = ∠4.

Seeing it with numbers

Look at triangle ABC below. Angle A measures 65° and angle C measures 30°. We claim that angle x must be 85°, so that the three angles add up to 180°.

To see why, draw a line through vertex B that is parallel to side AC.

• Side AB is a transversal of the two parallel lines, so angle a and the 65° angle at A are alternate interior angles. That makes angle a = 65°.
• Side BC is also a transversal, so angle b and the 30° angle at C are alternate interior angles. That makes angle b = 30°.
• Angles a, x and b sit side by side and form a straight line, so 65° + x + 30° = 180°.
• Solving: x = 180° − 95° = 85°.

This example shows that the theorem works for one triangle. It is not yet a proof, because a proof must work for every triangle.

The proof

Start with any triangle ABC, with angles x at A, c at B and y at C. Draw a line through B parallel to AC. This creates angles a and b on either side of angle c.

Since x, c and y are the three angles of triangle ABC, the angles of the triangle add up to 180°. Nothing in the proof depended on the size or shape of the triangle, so it works for every triangle. ∎

What follows from the theorem

• Each angle of an equilateral triangle measures 180° ÷ 3 = 60°.
• In a right triangle, the two acute angles add up to 180° − 90° = 90°.
• A triangle can have at most one right angle or obtuse angle, because two of them would already add up to 180° or more.

Practice problems

Pick an answer for each question. You'll see right away whether it's correct, with an explanation.