# Exterior Angles of Triangle: Theorem, Formula, Examples

TL;DR

An exterior angle of a triangle is formed by extending one side, and it equals the sum of the two non-adjacent (remote) interior angles. Each exterior angle is also supplementary to its adjacent interior angle, and the three exterior angles taken one per vertex sum to 360°. This article covers the exterior angle theorem, the formulas, a short proof, and six worked 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](/content/math/geometry/interior-angles/index.html) of a triangle, and it is one of the standard [properties of a triangle](/content/math/geometry/properties-of-a-triangle/index.html).

## The Exterior Angle Theorem

The headline result, the [exterior angle theorem](/content/math/geometry/exterior-angle-theorem/index.html), states:

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

In symbols, if the exterior angle at CCC is ∠ACD and the remote interior angles are ∠A and ∠B:

∠ACD=∠A+∠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](/content/math/geometry/triangle-sum-theorem/index.html):

∠A+∠B+∠ACB=180°

Second, the linear pair along the extended side:

∠ACB+∠ACD=180°

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

∠A+∠B+∠ACB=∠ACB+∠ACD

Subtract ∠ACB from both sides:

∠A+∠B=∠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:

exterior angle sum=360°

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

(180°−∠A)+(180°−∠B)+(180°−∠C)=540°−(∠A+∠B+∠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): exterior angle=180°−adjacent interior angle.
- **From the two remote interior angles** (the theorem): exterior angle=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](/content/math/geometry/alternate-angles/index.html) 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:

exterior angle=50°+70°

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:

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°+∠x=130°

∠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:

exterior angle=40°+75°=115°

_Way 2 — the linear pair._ First the third interior angle: 180°−40°−75°=65°. Then:

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.

## 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°.

### 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.

### 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 xxx, 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.