Triangle Angle Sum Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Triangle Angle Sum.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The three interior angles of any triangle always sum to exactly 180°, so knowing two angles determines the third.

Tear off the three corners of any paper triangle and line them up—they always form a straight line (180°). No matter how pointy or flat the triangle is, the angles always add up the same way, like three puzzle pieces that always complete a half-turn.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: This is a fundamental constraint on all triangles—knowing two angles immediately determines the third.

Common stuck point: This only applies to flat (Euclidean) geometry. On a sphere, triangle angles can sum to more than 180°.

Worked Examples

Example 1

easy
A triangle has angles 55° and 72°. Find the third angle.

Solution

  1. 1
    Step 1: Recall the Triangle Angle Sum Theorem: the three interior angles of any triangle sum to 180°.
  2. 2
    Step 2: Let the third angle be x. Then 55° + 72° + x = 180°.
  3. 3
    Step 3: Solve: x = 180° - 55° - 72° = 180° - 127° = 53°.

Answer

The third angle is 53°.
The Triangle Angle Sum Theorem states that the interior angles of every triangle add to exactly 180°. This holds for all triangles — acute, right, and obtuse. To find a missing angle, subtract the sum of the known angles from 180°.

Example 2

medium
In an isosceles triangle, the vertex angle is 40°. Find the two base angles.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Can a triangle have angles of 90°, 91°, and 1°? Justify your answer.

Example 2

hard
In \triangle ABC, \angle A = 2x + 10°, \angle B = 3x - 5°, \angle C = x + 15°. Find all three angles.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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