Angles in a triangle

Draw any triangle you like: tall and thin, short and wide, tiny or huge. Measure its three angles and add them up. You will always get the same number: 180°. The Greek mathematician Euclid wrote down a proof of this fact about 2,300 years ago, and it is still taught in schools today. It is one of the most useful facts in geometry. If you know two angles of a triangle, you can always find the third.

The rule

You can see why with a sheet of paper. Cut out a triangle. Tear off its three corners. Now put the three corners next to each other, with their points touching. They fit together exactly to make a straight line. A straight line is 180°, so the three angles together are 180°.

Finding a missing angle

When you know two angles, the third one is what is left over from 180°.

1. Add the two angles you know.
2. Subtract that total from 180°.
3. Check: all three angles together must make 180°.

You can also subtract the two angles one at a time: 180° − 45° = 135°, and then 135° − 68° = 67°. Both ways give the same answer. Use the one you find easier.

Triangles with a right angle

A right triangle has one right angle, which is 90°. In a drawing it is often marked with a small square in the corner. The right angle uses up half of the 180°, so the other two angles share the other 90°.

Equilateral triangles

An equilateral triangle has three equal sides. Its three angles are equal too. The 180° is shared out equally between three corners.

Isosceles triangles

An isosceles triangle has two sides of the same length. The two angles at the ends of the third side are equal. When the triangle stands on that third side, the two equal angles are the two bottom angles. They are called the base angles. The angle at the top can be different.

There are two kinds of question. Sometimes you know the top angle and need the bottom ones. Sometimes you know a bottom angle and need the top one.