Triangle ABC isosceles AC base, BD median. prove that triangles ABD and DBC are right-angled.

In order to prove that triangles ABD and DBC are right-angled, it is necessary to prove that <ADB = <CDB = 90 °. We will prove this in two ways.

1) In an isosceles triangle, the property of the median BD, lowered to the base of the AC from the vertex B: the median is the bisector, and the height. And since BD is the height, the angle is <ADB = <CDB = 90 °.

2) Let us prove that triangles ABD and CBD are equal.

These triangles are equal, since they have a common side BD.

Sides AB = BC, as sides of an isosceles triangle. And the sides are AD = CD, since the median bisects the base. The triangles are equal on three sides. This means that the angles are <ADB = <CDB = 90 °, and the triangles are right-angled.