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:
- A triangle has angles 33° and 91°. Find the third angle and classify the triangle.
- An equilateral triangle has one angle labelled (2x + 10)°. Find x.
- Two angles of a triangle are x and 3x. The third angle is 60°. Find x.
- An exterior angle of a triangle measures 135°. One of the non-adjacent interior angles is 80°. Find the other non-adjacent interior angle.
- 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°).