the isosceles-triangle theorem

An isosceles triangle has two equal sides, and the theorem reveals the consequence hidden in that symmetry: the two angles at the base — the angles opposite the equal sides — are themselves equal. In words students remember, 'the base angles of an isosceles triangle are equal.' The converse is also true and equally useful: if two angles of a triangle are equal, the sides opposite them are equal, so the triangle is isosceles.

Precisely: in triangle ABC with |AB| = |AC|, the base angles satisfy angle B = angle C. A clean proof draws the angle bisector from the apex A to a point M on BC. Then triangles ABM and ACM share AM, have |AB| = |AC| (given) and equal angles at A (the bisector), so they are congruent by SAS; CPCTC then gives angle B = angle C. The same bisector turns out to be the median and the altitude to the base as well — in an isosceles triangle these three special lines from the apex coincide. The medieval name pons asinorum, the 'bridge of asses', marks this as the first real test a geometry student crosses.

An honesty point worth stressing: equal base angles are a theorem, not part of the definition. The definition of isosceles is 'two equal sides'; the equality of the base angles is a proved consequence. Conflating the two — assuming the angles are equal 'by definition' — skips exactly the reasoning the theorem supplies.