Triangles: Classification, Angle Sum & Congruence

Triangles are the simplest polygons and the foundation of all geometry. Every polygon can be broken into triangles, so mastering triangle properties gives you power over all shapes. This lesson covers how to name them, their angle rules, and when two triangles are exactly the same.

Classification by Sides

Type	Sides	Key Feature
Scalene	All three sides different	All three angles different too
Isosceles	At least two sides equal	The two base angles are also equal
Equilateral	All three sides equal	All three angles are 60°

Classification by Angles

Type	Largest Angle
Acute triangle	All angles less than 90°
Right triangle	One angle equals 90°
Obtuse triangle	One angle greater than 90°

A triangle can have both classifications: for instance, a "right isosceles" triangle has a 90° angle and two equal legs (with 45° base angles).

Triangle Angle-Sum Property

The three interior angles of every triangle sum to exactly 180°.

This means if you know two angles, you can always find the third: subtract their sum from 180°.

Exterior Angle Theorem

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

Exterior angle = Sum of the two remote interior angles

Example 1 — Finding a Missing Angle

A triangle has angles of 48° and 73°. Find the third angle.

Sum of known angles: 48 + 73 = 121.

Third angle = 180 − 121 = 59°.

Classification: all angles < 90°, all sides likely different → acute scalene.

Example 2 — Exterior Angle

A triangle has interior angles of 35° and 80°. An exterior angle is adjacent to the third interior angle. Find the exterior angle.

Exterior angle = sum of remote interior angles = 35 + 80 = 115°.

Alternatively: third interior angle = 180 − 115 = 65°, and the exterior angle = 180 − 65 = 115°. Same answer.

Triangle Congruence

Two triangles are congruent if they have exactly the same shape and size (all corresponding sides and angles match). You do not need to check all six measurements. Any one of these shortcuts is enough:

Criterion	What You Need	Meaning
SSS	3 pairs of equal sides	Side-Side-Side
SAS	2 pairs of equal sides and the angle between them equal	Side-Angle-Side
ASA	2 pairs of equal angles and the side between them equal	Angle-Side-Angle

Example 3 — Identifying Congruence

Triangle PQR: PQ = 5 cm, QR = 8 cm, angle Q = 40°. Triangle XYZ: XY = 5 cm, YZ = 8 cm, angle Y = 40°. Are they congruent?

Two sides and the included angle match.

This fits SAS, so the triangles are congruent.

Common Mistake

SSA is NOT a valid congruence criterion. Knowing two sides and a non-included angle can sometimes produce two different triangles (the "ambiguous case"). Only use SSS, SAS, or ASA.

Equilateral Shortcut

If a triangle is equilateral, every angle is 60°—no calculation needed. And any two equilateral triangles with the same side length are congruent by SSS.

Practice Problems

1. A triangle has angles 60°, 60°, 60°. Classify it by sides and angles.

Show Solution

Equilateral (all sides equal) and acute (all angles < 90°).

2. Find the missing angle: 90°, 34°, ?

Show Solution

180 − 90 − 34 = 56°. This is a right scalene triangle.

3. The exterior angle of a triangle is 130°. One remote interior angle is 55°. Find the other remote interior angle.

Show Solution

130 − 55 = 75°.

4. Triangle ABC has sides 7, 7, 10. Triangle DEF has sides 7, 7, 10. Are they congruent? State the criterion.

Show Solution

Yes, by SSS (all three pairs of sides are equal).

5. An isosceles triangle has a vertex angle of 110°. Find each base angle.

Show Solution

Base angles are equal. (180 − 110) / 2 = 70 / 2 = 35° each.

Lesson Summary

Triangles are classified by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse).
The three interior angles always sum to 180°.
An exterior angle equals the sum of the two remote interior angles.
Congruence can be proven with SSS, SAS, or ASA (never SSA alone).