Classifying Triangles

Overview

Every triangle can be classified in two independent ways: by its angles and by its sides. The angle classification looks at whether the triangle has acute, right, or obtuse angles, while the side classification counts how many sides are congruent. Special vocabulary names the important parts of right triangles and isosceles triangles, and in any triangle each side is opposite a specific angle — a pairing that matters throughout the unit.

Vocabulary

Triangle: A three-sided polygon. A triangle has three sides, which are line segments, and three interior angles; the points where the sides meet are called vertices. The three interior angles of a triangle always add to 180°.
Acute Triangle: A triangle with three acute angles — every angle measures less than 90°.
Right Triangle: A triangle with exactly one right angle (90°). The other two angles must be acute, and they are complementary (their measures add to 90°).
Obtuse Triangle: A triangle with exactly one obtuse angle — one angle measuring between 90° and 180°. The other two angles must be acute.
Equiangular Triangle: A triangle in which all three angles are congruent, so each angle measures 60°.
Scalene Triangle: A triangle in which all three sides have different lengths — no sides are congruent.
Isosceles Triangle: A triangle with at least two congruent sides. The two congruent sides are called legs.
Equilateral Triangle: A triangle in which all three sides are congruent. An equilateral triangle is also equiangular.

Main Ideas

Classify by angles: acute (three acute angles), right (one right angle), obtuse (one obtuse angle), or equiangular (three congruent angles).
Classify by sides: scalene (no sides congruent), isosceles (at least two sides congruent), or equilateral (three sides congruent). A complete classification names both.
Congruent parts come in matching counts: a triangle with three congruent angles has three congruent sides, a triangle with two congruent angles has two congruent sides, and a triangle with no congruent angles has no congruent sides.
In a right triangle, the side opposite the right angle is the hypotenuse, and the two sides that form the right angle are the legs.
In an isosceles triangle, the angle between the two legs is the vertex angle, the side opposite it is the base, and the two base angles are congruent.
Each vertex of a triangle is opposite the side that does not touch it: in △ABC, vertex A is opposite side BC, and side AB is opposite ∠C.