Triangle Classification and the Angle Sum Theorem

By the end of this lesson, you will be able to classify triangles by their angles and sides, apply the Triangle Angle-Sum Theorem (interior angles sum to 180°), and use the Exterior Angle Theorem.

The Big Idea

The Triangle Angle-Sum Theorem — that every triangle's interior angles sum to 180° — is one of the most important results in Euclidean geometry. It's not obvious, it requires proof, and it has dramatic consequences: once you know two angles of a triangle, the third is determined completely.

Classifying by Angles

Type	Definition
Acute triangle	All three angles acute (< 90°)
Right triangle	One angle exactly 90°
Obtuse triangle	One angle obtuse (90°–180°)

A triangle can have at most one right or obtuse angle (since the sum of all three is 180°, two angles ≥ 90° would exceed 180°).

Classifying by Sides

Type	Definition
Scalene	No congruent sides
Isosceles	At least two congruent sides
Equilateral	All three sides congruent

An equilateral triangle is always also equiangular (all angles = 60°) — and vice versa.

Triangle Angle-Sum Theorem

The sum of the interior angles of any triangle is 180°.

$$m\angle A + m\angle B + m\angle C = 180°$$

Proof using parallel lines:

Draw a line through vertex $A$ parallel to $\overline{BC}$. This creates alternate interior angles with $\angle B$ and $\angle C$. The three angles along the top (the two alternate interior angles plus $\angle A$) form a straight angle = 180°. Since alternate interior angles with a parallel line are congruent, the sum equals the sum of the triangle's angles.

Example 1

Find the missing angle: $\angle A = 47°$, $\angle B = 83°$.

$$\angle C = 180 - 47 - 83 = 50°$$

Example 2 — Algebra

In a triangle, angles are $x°$, $(2x + 10)°$, and $(x - 10)°$. Find all angles.