Isosceles and Equilateral Triangles

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

The converse is also true

Two equal angles force the sides opposite them to be equal.

Equilateral

All three sides equal makes all three angles equal, and 180 ÷ 3 is 60.

Why it works

The perpendicular from the apex splits the triangle into two congruent halves.

Equal sides force equal angles

In an isosceles triangle the angles opposite the equal sides are equal. The proof draws the bisector of the apex angle, producing two congruent triangles by SAS. It is a short proof and a good first one to reconstruct from scratch.

The converse also holds

If two angles of a triangle are equal, the sides opposite them are equal. Both directions are true here, which is not automatic — many geometric statements have false converses, so each direction needs its own proof.

Equilateral means equiangular

A triangle with three equal sides has three equal angles, each 60°, since they must total 180°. Applying the isosceles result three times gives this immediately, which is an example of building results on results.

Where it gets used

Radii of a circle are equal, so any triangle formed by two radii is isosceles. That single observation is behind the inscribed angle theorem and much of circle geometry. Spotting hidden isosceles triangles is a valuable habit.