The Angle Sum of a Triangle

Roof trusses, bridge girders, sat-nav fixes and a snooker player's bank shot all lean on one unshakeable fact about triangles. Because the three corners of any triangle always add up to the same amount, knowing two of the angles instantly hands you the third.

Draw any triangle you like — tall, squat, lopsided, huge or tiny — and add up its three inside angles. You always get exactly the same total:

\angle A + \angle B + \angle C = 180^\circ

180^\circ is a straight angle — a flat half-turn, the amount you swing through if you spin from looking one way to looking the exact opposite way. No matter how you stretch or squash a triangle, its three corners always add up to that same flat half-turn. This one fact is the workhorse of triangle geometry, and the reason it is true comes straight from the alternate interior angles of parallel lines.

In every triangle, the three interior angles add up to a straight angle:

the sum is always \angle A + \angle B + \angle C = 180^\circ, whatever the triangle's shape or size;
so any two angles determine the third — it is 180^\circ minus the other two;
it follows from alternate interior angles: a line through the apex parallel to the base turns the three angles into angles on one straight line.

Tear off the corners and see it

Here is a way to see it with your hands — no maths needed. Cut out any paper triangle. Now tear off all three corners. Lay the three torn corners down side by side, with their tips all touching the same point and their edges butting together, along the edge of a ruler:

Every single time, the three corners fit together perfectly to make one straight line — and a straight line is a half-turn, which is angles on a straight line adding to 180^\circ. Try it with a tall thin triangle and again with a short fat one: the corners come out different sizes, but they always close up into the very same flat straight line. That is the angle sum, right there on your desk.

slice of pizza

A thin slice of pizza is a triangle with a very sharp point where it was cut from the middle — maybe only 20^\circ — and two wider corners at the crust. The tiny tip and the two bigger corners still add up to 180^\circ. Cut a fatter slice and the tip grows while the crust corners shrink, but the three always trade fairly so the total never changes. Big slice or small slice: always a flat half-turn.

The proof, one step at a time

The trick is to draw a single extra line. Step through it: a line through the apex parallel to the base turns the triangle's angles into three angles on a straight line.

Because the alternate angles are equal, the angles on the line are the very same \angle A and \angle B from the triangle — and a straight line is 180^\circ. So \angle A + \angle C + \angle B = 180^\circ, which is exactly the triangle's three angles.

Using it: find the missing angle

Once you know two angles of a triangle, the third has nowhere to hide — it is whatever is left over after you take the two you know away from 180^\circ:

\text{missing angle} = 180^\circ - (\text{angle one}) - (\text{angle two})

For example, a triangle with angles 60^\circ and 50^\circ has a third angle of 180^\circ - 60^\circ - 50^\circ = 70^\circ. Below is a fresh triangle with two angles marked and one shown as ?. Work out the missing angle, then press Next to check, or Refresh for a new one.

Worked examples

Every one is the same move — add the two you know, then take that away from 180°.

Two angles given. A triangle has angles 40^\circ and 75^\circ. Together that is 40^\circ + 75^\circ = 115^\circ, so the third angle is 180^\circ - 115^\circ = 65^\circ.
A right-angled triangle. One corner is a square right angle, 90^\circ, and another is 25^\circ. The last angle is 180^\circ - 90^\circ - 25^\circ = 65^\circ. (In a right-angled triangle the other two angles always add to 90^\circ.)
An isosceles triangle. The pointy top angle is 40^\circ, and the two base angles are equal. They share what is left: 180^\circ - 40^\circ = 140^\circ split in two, so each base angle is 140^\circ \div 2 = 70^\circ.

One thing to watch

The most common slips with the triangle angle sum:

The three angles total 180^\circ, not 360^\circ. 360^\circ is a full turn (and the angle sum of a four-sided shape) — a triangle is only a half-turn.
It works for every triangle — a giant one, a tiny one, a long thin sliver. Making a triangle bigger does not make its angles bigger; the total stays 180^\circ.
To find a missing angle, subtract both known angles from 180^\circ — not just one of them.

Look at the five spikes of a star. Each spike is a sharp little triangle, with a small pointy tip and two corners at the base. The tip might be only 36^\circ, but its three angles still add up to the same 180^\circ as the broad triangle of a mountain or a sail. Sharp or wide, near or far, every triangle keeps the same total — that is what makes the rule so handy.

Practise: chase the angles