## What You Will Learn

By the end of this page, you will know that the three interior angles of any triangle sum to 180°, be able to find any missing angle given the other two, classify triangles by their angles, and use the exterior angle theorem.

## Theory

### The Triangle Angle Sum Property

The three interior angles of any triangle always add up to exactly 180°.

∠A + ∠B + ∠C = 180°

This means: if you know any two angles, you can always find the third by subtracting from 180°.

**Why 180°?** If you draw a straight line through a triangle's vertex parallel to the opposite side, the three angles at that vertex form a straight line — which is exactly 180°.

### Types of Triangles by Angles

| Type                | Angle property                               |
|---------------------|---------------------------------------------|
| Acute triangle      | All three angles < 90°                       |
| Right triangle      | Exactly one angle = 90°                      |
| Obtuse triangle     | Exactly one angle > 90°                       |
| Equilateral triangle| All angles = 60°                            |

### The Exterior Angle Theorem

An **exterior angle** is formed by extending one side of a triangle beyond the vertex. It equals the sum of the two **non-adjacent interior angles** (also called remote interior angles).

Exterior angle = ∠A + ∠B

where ∠A and ∠B are the two interior angles not next to the exterior angle.

### Special Triangle Properties

**Isosceles triangle:** two equal sides → the two **base angles** (angles opposite the equal sides) are also equal.

**Equilateral triangle:** all three sides equal → all three angles are 60°.

**Right triangle:** one angle is 90° → the other two angles are complementary (they add up to 90°).

## Worked Examples

### Example 1: Finding a Missing Angle

**Problem:** A triangle has angles of 47° and 68°. Find the third angle.

**Step 1:** Use the angle sum property.
∠C = 180° − 47° − 68°

**Step 2:** Calculate.
∠C = 180° − 115° = 65°

**Answer:** The third angle is 65°.

### Example 2: Isosceles Triangle

**Problem:** An isosceles triangle has a vertex angle of 40°. Find the two base angles.

**Step 1:** The two base angles are equal. Let each base angle = x.

**Step 2:** Apply the angle sum.
40° + x + x = 180° → 2x = 140° → x = 70°

**Answer:** Each base angle is 70°.

### Example 3: Using the Exterior Angle Theorem

**Problem:** Two interior angles of a triangle are 52° and 74°. Find the exterior angle at the third vertex.

**Step 1:** The exterior angle equals the sum of the two non-adjacent interior angles.
Exterior angle = 52° + 74° = 126°

**Answer:** The exterior angle is 126°.

### Example 4: Algebraic Angle Problem

**Problem:** The angles of a triangle are x, 2x, and 3x. Find each angle and classify the triangle.

**Step 1:** Set up the equation.
x + 2x + 3x = 180° → 6x = 180° → x = 30°

**Step 2:** Find each angle.
x = 30°, 2x = 60°, 3x = 90°

**Step 3:** Classify. One angle is exactly 90° — this is a **right triangle**.

**Answer:** Angles are 30°, 60°, 90° — a right triangle.

## Common Mistakes

**Mistake 1: Setting Angles Equal to 90° Instead of 180°**

**Correct:** The angles of a triangle sum to 180°, not 90°.

**Mistake 2: Using the Exterior Angle as an Interior Angle**

**Correct:** The exterior angle is outside the triangle. The interior angle at that vertex is 180° − exterior angle.

**Mistake 3: Assuming All Triangles Can Have Two Obtuse Angles**

**Correct:** A triangle can have at most one obtuse angle.

## Practice Problems

Try these on your own before checking the answers:

1. A triangle has angles 33° and 91°. Find the third angle and classify the triangle.
2. An equilateral triangle has one angle labelled (2x + 10)°. Find x.
3. Two angles of a triangle are x and 3x. The third angle is 60°. Find x.
4. An exterior angle of a triangle measures 135°. One of the non-adjacent interior angles is 80°. Find the other non-adjacent interior angle.
5. Can a triangle have angles of 90°, 90°, and 0°? Explain.

## Summary

- The three interior angles of any triangle sum to 180°.
- To find a missing angle: subtract the known angles from 180°.
- Exterior angle theorem: exterior angle = sum of the two non-adjacent interior angles.
- Triangle types: acute (all < 90°), right (one = 90°), obtuse (one > 90°), equilateral (all = 60°).