Isosceles & Equilateral Triangles

Base angles are equal; an equilateral triangle is all 60°.

The idea

A triangle is isosceles when two of its sides are equal, and equilateral when all three are. In an isosceles triangle, the angle between the two equal sides is the vertex angle; the third side is the base, and the two angles at its ends, each opposite one of the equal sides, are the base angles.

Theorem (Base Angles Theorem).

In an isosceles triangle, the two base angles are equal: the angles opposite the two equal sides are equal.

Combined with the fact that a triangle's angles sum to $180°$, one angle determines the other two. From the vertex angle $v$, each base angle is $\frac{180° - v}{2}$; from a base angle $b$, the other base angle is also $b$, and the vertex angle is $180° - 2b$.

In an equilateral triangle any two sides are equal, so any two angles are equal, and all three are the same. Three equal angles summing to $180°$ make each angle $60°$.

Ways to work on it

• Walkthrough. The Base Angles Theorem, vertex versus base angles, and equilateral 60°.
• Proof. Why the base angles are equal: SSS congruence of the two halves.
• Practice. Convert between vertex and base angles.
• Hardest. Solve for the angles when they are given algebraically.

Not sure where to start? Take the ten-question placement test.