In triangle ABC, angles A and C are equal, BD is the height of the triangle. Prove that triangles ABD and CBD are equal.

If two angles in a triangle are equal, then this triangle is called isosceles. Angles A and C are equal by condition, which means that triangle ABC is isosceles.
The height BD drawn from the vertex B is also the bisector and the median, which means that AD = CD.
Consider triangles ABD and CBD, in them:
AB = BC (sides of an isosceles triangle);
AD = CD (BD – median);
BD – general.
The triangles are equal on three sides. Q.E.D.