Triangle Angle Sum Formula

The Formula

\angle A + \angle B + \angle C = 180°

When to use: 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.

Quick Example

A triangle with angles 50° and 70°: \angle C = 180° - 50° - 70° = 60°

Notation

\angle A, \angle B, \angle C are the three interior angles of \triangle ABC

What This Formula Means

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.

Formal View

In Euclidean geometry (\mathbb{R}^2): \forall\,\triangle ABC, m(\angle A) + m(\angle B) + m(\angle C) = \pi rad = 180°; equivalently, the defect \delta = \pi - (\angle A + \angle B + \angle C) = 0 (nonzero on curved surfaces)

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.

Common Mistakes

  • Applying the rule to non-triangular polygons (quadrilaterals sum to 360°, not 180°)
  • Arithmetic errors when subtracting to find the missing angle
  • Confusing interior angles with exterior angles

Why This Formula Matters

The most-used fact in geometry. It lets you find missing angles and is the basis for proving many other theorems.

Frequently Asked Questions

What is the Triangle Angle Sum formula?

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

How do you use the Triangle Angle Sum formula?

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.

What do the symbols mean in the Triangle Angle Sum formula?

\angle A, \angle B, \angle C are the three interior angles of \triangle ABC

Why is the Triangle Angle Sum formula important in Math?

The most-used fact in geometry. It lets you find missing angles and is the basis for proving many other theorems.

What do students get wrong about Triangle Angle Sum?

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

What should I learn before the Triangle Angle Sum formula?

Before studying the Triangle Angle Sum formula, you should understand: triangles, angles, addition.

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