Exterior Angles of Triangle: Theorem, Formula, Examples

What Is an Exterior Angle of a Triangle?

An exterior angle of a triangle is the angle formed between one side of the triangle and the extension of an adjacent side. At each vertex you can extend a side in one of two directions, so a triangle has six exterior angles in total — but we usually work with one per vertex, three in all.

Every exterior angle pairs with the interior angle right beside it. Together they sit on a straight line, so they form a linear pair and add to 180°. The two interior angles away from the exterior angle — the ones it does not touch — are called the remote interior angles (or opposite interior angles).

This builds directly on the interior angles of a triangle, and it is one of the standard properties of a triangle.

The Exterior Angle Theorem

The headline result, the exterior angle theorem, states:

An exterior angle of a triangle equals the sum of its two remote interior angles.

In symbols, if the exterior angle at $C$ is $\angle ACD$ and the remote interior angles are $\angle A$ and $\angle B$:

$$\angle ACD = \angle A + \angle B$$

There is also an exterior angle inequality: any exterior angle is strictly greater than either one of its remote interior angles (since it equals their sum, it must beat each one alone).

A Short Proof

The proof uses two facts you already know, side by side.

First, the triangle sum theorem:

$$\angle A + \angle B + \angle ACB = 180°$$

Second, the linear pair along the extended side:

$$\angle ACB + \angle ACD = 180°$$

Both right-hand sides are 180°, so set the left-hand sides equal:

$$\angle A + \angle B + \angle ACB = \angle ACB + \angle ACD$$

Subtract $\angle ACB$ from both sides:

$$\angle A + \angle B = \angle ACD$$

The exterior angle equals the sum of the two remote interior angles.

The Sum of Exterior Angles of a Triangle

Take one exterior angle at each of the three vertices and add them:

$$\text{exterior angle sum} = 360°$$

Here is why. Each exterior angle is $180°$ minus its adjacent interior angle. Add the three:

$$(180° - \angle A) + (180° - \angle B) + (180° - \angle C)$$

$$= 540° - (\angle A + \angle B + \angle C)$$

$$= 540° - 180° = 360°$$

This 360° result is not special to triangles — the exterior angles of any convex polygon, one per vertex, total 360°.

Exterior Angle of a Triangle Formulas

A short opener: there are two ways to find an exterior angle, depending on what you know.

• From the adjacent interior angle (linear pair): $\text{exterior angle} = 180° - \text{adjacent interior angle}$.

• From the two remote interior angles (the theorem): $\text{exterior angle} = \text{sum of the two remote interior angles}$.

• Sum of all three (one per vertex): always $360°$.

The link to parallel-line geometry runs deep — when a transversal cuts parallel lines, the alternate angles it creates are the same machinery that makes the exterior angle theorem work.

Examples of Exterior Angles of Triangle

Example 1

A triangle has remote interior angles of 50° and 70°. Find the exterior angle at the third vertex.

By the exterior angle theorem:

$$\text{exterior angle} = 50° + 70°$$

$$\text{exterior angle} = 120°$$

Final answer: 120°.

Example 2 (the most common slip, walked through)

The interior angle adjacent to an exterior angle is 56°. The exterior angle is reported as 56°. Is that right?

Wrong attempt. A reflex is to copy the adjacent interior angle straight across: "the angle next to it is 56°, so the exterior angle is 56° too."

Why it breaks. The exterior angle and its adjacent interior angle form a straight line — they are supplementary, not equal. Setting them equal would mean two angles on a straight line each measuring 56°, totalling only 112°, not 180°.

Correct. Use the linear pair:

$$\text{exterior angle} = 180° - 56° = 124°$$

Final answer: 124° — the adjacent angle is the exterior angle's supplement, never its twin.

Example 3

An exterior angle of a triangle is 130°. One remote interior angle is 85°. Find the other remote interior angle.

By the theorem, the two remote interior angles sum to the exterior angle:

$$85° + \angle x = 130°$$

$$\angle x = 130° - 85° = 45°$$

Final answer: 45°.

Example 4

Two interior angles of a triangle are 40° and 75°. Find the exterior angle at the third vertex two ways.

Way 1 — the theorem. The remote interior angles for the third vertex are the two given angles:

$$\text{exterior angle} = 40° + 75° = 115°$$

Way 2 — the linear pair. First the third interior angle: $180° - 40° - 75° = 65°$. Then:

$$\text{exterior angle} = 180° - 65° = 115°$$

Final answer: 115° (both routes agree).

Example 5

The exterior angles of a triangle, one per vertex, are $(2x)°$, $(3x)°$, and $(4x)°$. Find $x$.

The three exterior angles sum to 360°:

$$2x + 3x + 4x = 360$$

$$9x = 360$$

$$x = 40$$

Final answer: $x = 40$, so the exterior angles are 80°, 120°, and 160°.

Example 6

In a triangle, an exterior angle equals 3 times one remote interior angle, and the other remote interior angle is 40°. Find the exterior angle.

Let the first remote interior angle be $r$. The exterior angle is $3r$, and by the theorem it equals the sum of the remotes:

$$3r = r + 40$$

$$2r = 40$$

$$r = 20$$

So the exterior angle is $3r = 60°$.

Final answer: 60°.

Why the Exterior Angle Theorem Matters

"An exterior angle is greater than either remote interior angle."

That inequality is Proposition 16 of Book I of Euclid's Elements — and Euclid proved it before he had the full angle-sum result, because it was the load-bearing step for proving lines parallel. The exterior angle is a measuring tool: it lets you pin down two unknown angles from one you can actually see.

Where it earns its keep:

• Navigation and bearings. A ship turning at a waypoint sweeps through the exterior angle of its course triangle; the turn angle is the exterior angle, and it equals the two remote heading changes combined.

• Roof and frame geometry. A carpenter reading the splay where a rafter meets an extended ridge is reading an exterior angle — far easier to measure on-site than the tight interior corner.

• Polygon angle sums. Because every convex polygon's exterior angles total 360°, you can find any regular polygon's interior angle in one step: each exterior angle is $360°/n$, so each interior angle is $180° - 360°/n$.

This last use connects exterior angles straight to the types of triangle and beyond, to every polygon a student will meet later.

Where Students Trip Up on Exterior Angles

Mistake 1: Setting the exterior angle equal to the adjacent interior angle

Where it slips in: Reading an exterior angle off a diagram with the adjacent interior angle marked.

Don't do this: Copy the adjacent interior angle as the exterior angle.

The correct way: The exterior angle and the adjacent interior angle are supplementary — subtract from 180°. They are equal only in the degenerate case where both are 90°.

The first-instinct error is treating "the angle right next to it" as the answer. The fix is to always ask which two angles add to 180° here — the adjacent pair sits on a straight line, so it is them.

Mistake 2: Adding all three interior angles instead of just the two remote ones

Where it slips in: Applying the exterior angle theorem.

Don't do this: Set the exterior angle equal to the sum of all three interior angles (which would give 180°, always wrong).

The correct way: The exterior angle equals only the two remote interior angles — the two it does not touch. Leave out the adjacent one.

The second-guesser does this most: they correctly recall "exterior equals a sum of interior angles" but lose track of which ones, so they sweep in all three to be safe. Name the adjacent angle first and exclude it; the remaining two are the remotes.

Mistake 3: Treating all six exterior angles as distinct when summing to 360°

Where it slips in: Computing the exterior angle sum.

Don't do this: Add all six exterior angles and expect 360°.

The correct way: The 360° sum uses one exterior angle per vertex — three of them. The other three are vertically opposite and equal, so adding all six gives 720°.

Key Takeaways

• An exterior angle of a triangle is formed by extending a side and equals the sum of the two remote interior angles.

• Each exterior angle is supplementary to its adjacent interior angle (they add to 180°).

• The three exterior angles, one per vertex, always sum to 360°.

• The proof combines the triangle sum (180°) with the straight-line linear pair (180°).

• The most common error is confusing the exterior angle with its adjacent interior angle — they are supplements, not equals.

A Practical Next Step

Practice these problems to solidify your understanding. Question 1: Remote interior angles 35° and 95° — find the exterior angle. Question 2: An exterior angle is 140° with one remote interior 65° — find the other. Question 3: Three exterior angles are $x$, $2x$, and $3x$ — find each. If you get stuck on Question 1, return to "The Exterior Angle Theorem" and add only the two angles the exterior angle does not touch.

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