Isosceles Triangles

An isosceles triangle has two equal sides. Its real charm is what happens at the corners: the two angles sitting opposite those equal sides — the base angles — are always exactly equal.

\text{two equal sides} \;\Longrightarrow\; \text{two equal base angles}

And it works both ways (the converse): if two angles of a triangle are equal, then the sides opposite them are equal too, so the triangle must be isosceles.

In a triangle with two equal sides meeting at the apex:

the two base angles (opposite the equal sides) are equal — \angle A = \angle B;
the line of symmetry from the apex bisects the base and bisects the apex angle, meeting the base at right angles;
with the angle sum, one angle gives the rest: an apex of \alpha makes each base angle \dfrac{180^\circ - \alpha}{2}.

Why it works

Fold the triangle along the line from the apex to the middle of the base. The two equal sides land on each other — so the two base angles must match. Step through it.

In symbols: the apex angle \alpha plus the two equal base angles \beta make 180^\circ, so 2\beta = 180^\circ - \alpha and \beta = \dfrac{180^\circ - \alpha}{2}.