Exterior Angles — Theorem & Sum of Exterior Angles

Exterior angles are the angles outside a shape that appear when you extend a side. The key facts are simple: in a triangle, an exterior angle equals the sum of the two remote interior angles, and in any polygon, one exterior angle at each vertex adds to $360^\circ$ if you measure them in the same direction.

If you only remember one shortcut, remember this: an interior angle and its adjacent exterior angle always add to $180^\circ$.

What An Exterior Angle Means

Take a polygon and extend one side past a vertex. The angle between that extension and the next side is an exterior angle.

That exterior angle sits next to the interior angle at the same vertex, so the two form a straight line:

$$\text{interior angle} + \text{adjacent exterior angle} = 180^\circ$$

This relation is often the fastest way to switch between interior and exterior angles.

How The Triangle Exterior Angle Theorem Works

For a triangle, an exterior angle equals the sum of the two interior angles that are not next to it. These are called the remote interior angles.

If the remote interior angles are $a$ and $b$, and the exterior angle is $e$, then

$$e = a + b$$

This only works with the two remote interior angles. The interior angle next to the exterior angle is not part of the theorem.

Why Exterior Angles Of A Polygon Add To $360^\circ$

If you take one exterior angle at each vertex of a polygon and measure them in the same turning direction, the total is always

$$360^\circ$$

This is true for triangles, quadrilaterals, pentagons, and larger polygons. A useful way to picture it is walking around the shape: the exterior angles track your total turn, and one full trip brings you back to your starting direction, which is a full turn of $360^\circ$.

For a regular polygon, all exterior angles are equal, so each one is

$$\frac{360^\circ}{n}$$

where $n$ is the number of sides.

Worked Example: Using The Triangle Exterior Angle Theorem

In a triangle, two remote interior angles are $48^\circ$ and $67^\circ$. Find the exterior angle at the third vertex.

Use the theorem directly:

$$e = 48^\circ + 67^\circ = 115^\circ$$

So the exterior angle is $115^\circ$.

If you also want the interior angle next to it, use the straight-line relationship:

$$\text{adjacent interior angle} = 180^\circ - 115^\circ = 65^\circ$$

This example shows the two most common moves:

1. Add the two remote interior angles to get the exterior angle.
2. Subtract from $180^\circ$ if the problem also asks for the adjacent interior angle.

For a regular polygon, the workflow is different: first use $360^\circ / n$ to find one exterior angle, then subtract from $180^\circ$ if you need the interior angle.

Common Exterior Angle Mistakes

Using The Wrong Angles In The Triangle Theorem

In a triangle, the exterior angle equals the sum of the two remote interior angles, not the interior angle next to it.

Adding More Than One Exterior Angle At A Vertex

The $360^\circ$ rule uses one exterior angle per vertex. If you count extra angles or mix different outside angles at the same corner, the total will not match the theorem.

Ignoring The Direction Condition

For polygons, use one exterior angle at each vertex and measure them consistently as you move around the shape. That is what makes the total represent one full turn.

Assuming Every Polygon Has Equal Exterior Angles

Only regular polygons have equal exterior angles. Irregular polygons still have a total exterior-angle sum of $360^\circ$, but the individual angles can be different.

When You Use Exterior Angles

Exterior angles show up in triangle proofs, polygon angle questions, and regular polygon problems. They are especially useful when you need an unknown angle quickly without solving every angle in the figure first.

They also connect geometry to turning. That is why the polygon sum is stable and easy to remember.

Try A Similar Problem

Try a regular decagon next. First find one exterior angle using $360^\circ / 10$, then find the adjacent interior angle.

If you want to check your reasoning step by step, compare your work with a solver after you finish and see whether you used the exterior-angle sum or the straight-line relationship at the right moment.