Triangle Classification

Intuition and core definition

Triangles are classified independently by side lengths and angle measures. By sides: scalene has no equal sides, isosceles at least two, equilateral three. By angles: acute has three acute angles, right has one right angle, and obtuse has one obtuse angle.

Notation, language, and conditions

Matching tick marks denote congruent sides and arc marks congruent angles. An equilateral triangle is also equiangular with each angle $60^\circ$, hence acute and isosceles under the inclusive “at least two” definition. Side lengths must satisfy the strict triangle inequality: the sum of any two exceeds the third.

Why this idea matters

Triangles can be classified independently by side equality and angle size, so one triangle carries one label from each system.

A dependable method

1. Verify three lengths can form a triangle.
2. Compare side lengths or tick marks for the side classification.
3. Use given angles or calculate missing angles from $180^\circ$.
4. Classify by the largest angle: below, equal to, or above $90^\circ$.
5. State both classifications when information permits.

Worked example

Common mistakes