GeoProof — Isosceles Triangle Theorem explorer

AB = AC ⟺ ∠B = ∠C

The Isosceles Triangle Theorem

An isosceles triangle has two equal sides. The theorem says the angles opposite those equal sides — the base angles — are equal:

if AB = AC then ∠B = ∠C (and the converse)

Here AB and AC are the legs from the apex A; ∠B and ∠C are the base angles. Lock the triangle isosceles and drag — the base angles stay equal however tall or flat it gets.

The converse

It also runs the other way: if two angles of a triangle are equal, the sides opposite them are equal, so the triangle is isosceles. Equal sides and equal base angles always come together.

Why it's true

Drop the line from the apex A to the midpoint of BC. It splits the triangle into two pieces with the same three side lengths (AB = AC, the shared median, and equal half-bases), so the two pieces are congruent — making ∠B and ∠C equal. For an isosceles triangle this line is at once the median, the altitude, the angle bisector from A, and the axis of symmetry.

Special case

If all three sides are equal (equilateral), all three angles are equal — each 60°.

A bit of history

This is Proposition 5 of Book I of Euclid's Elements, famously nicknamed the pons asinorum ("bridge of asses") — an early test of a geometry student.