The Exterior Angle of a Triangle

Imagine walking all the way around a triangular park, tracing the fence. Down one side you go in a straight line — then you reach a corner and have to turn before setting off down the next side. The amount you swing round at that corner is the exterior angle of the triangle there.

You could measure that turn the slow way (walk the interior angle, subtract from a straight line) — but there is a lovely shortcut hiding in every triangle. The exterior angle at a corner is exactly the sum of the two angles at the far corners — the two corners the turn is not touching:

\text{exterior angle} = \text{the two remote interior angles, added}

("Remote" just means far away — the two corners the exterior angle is not at.) It drops straight out of the angle sum of a triangle, and once you spot it you can find angles in your head that would otherwise take three steps.

Extend one side of a triangle to make an exterior angle at a vertex. Then:

the exterior angle equals the sum of the two remote interior angles — \angle\text{ext} = \angle A + \angle C for the exterior angle at B;
it also makes a straight line with the interior angle next to it, so \angle\text{ext} + \angle B = 180^\circ;
so an exterior angle is always bigger than either remote interior angle.

Why it works

Two facts you already know do all the work: a straight line is 180^\circ, and a triangle's angles add to 180^\circ. Step through it.

In symbols: the interior angle at B is 180^\circ - (\angle A + \angle C) by the angle sum, and the exterior angle is 180^\circ - \angle B by the straight line — so the exterior angle is \angle A + \angle C. The two 180^\circs cancel, leaving the shortcut behind.

Worked examples

1) Two ways, same answer. A triangle has interior angles of 50^\circ and 70^\circ at two corners. Find the exterior angle at the third corner.

The long way (via the straight line). First the third interior angle: 180^\circ - 50^\circ - 70^\circ = 60^\circ. The exterior angle sits on a straight line with it, so 180^\circ - 60^\circ = 120^\circ.
The shortcut. The exterior angle equals the two remote interior angles added: 50^\circ + 70^\circ = 120^\circ.

Both give 120^\circ — but the shortcut skipped a whole step. The two 180^\circs were quietly cancelling each other the whole time.

2) Working backwards. An exterior angle is 115^\circ, and one of its remote interior angles is 48^\circ. What is the other remote interior angle?

The two remote angles add to the exterior angle, so the missing one is 115^\circ - 48^\circ = 67^\circ. (Check: the interior angle next to the exterior one is 180^\circ - 115^\circ = 65^\circ, and 65^\circ + 48^\circ + 67^\circ = 180^\circ — the whole triangle adds up.)

3) Which is quicker? A triangle has angles 38^\circ and 84^\circ; find the exterior angle at the third corner. The long way is 180^\circ - 38^\circ - 84^\circ = 58^\circ, then 180^\circ - 58^\circ = 122^\circ — two subtractions. The shortcut is one addition: 38^\circ + 84^\circ = 122^\circ. Same answer, half the work.

The exterior angle equals the sum of the two remote (non-adjacent) interior angles — the two far corners. Three traps catch people out:

Not all three. Adding all three interior angles gives 180^\circ every time — that's the angle sum, not the exterior angle.
Not the adjacent one. The interior angle right next to the exterior angle is the one you leave out. It has a different relationship: the exterior angle and its adjacent interior angle are supplementary — they sit on a straight line and add to 180^\circ, they are not equal.
So two different rules live at one corner. "Adjacent → add to 180°" and "remote → add up to the exterior angle." Pick the pair that are far from the turn.

Keep walking round the triangular park, turning by the exterior angle at each of the three corners, until you're back where you started facing the way you began. How much did you turn altogether? A full circle — 360^\circ. That's true for a triangle, a square, a hexagon, a wobbly 20-sided blob: the exterior angles of any polygon always add up to 360^\circ, because one lap of turning is one full spin.

That's why this little angle matters far beyond triangles — you'll use it to crack every regular shape when you meet the exterior angles of polygons.

Practise: chase the angles

A triangle with one side extended. Fill in every angle you can — use the exterior-angle rule, angles on a straight line, and the triangle's angle sum — ending with the highlighted one. Refresh for a new triangle; Check explains each step.