The bisector BD is drawn in an isosceles triangle ABC with base AC. Prove that triangle ABD is equal to triangle CBD

Since, by condition, ВD is the bisector of the angle, then the angle ABD = CBD.

In an isosceles triangle, the bisector drawn to the base of the triangle is also the height and median of the triangle, and therefore AD = CD, and the angles CDB and CBD are straight.

Then triangles ABD and LВD are rectangular.

The side BD is common for the triangles.

Then the triangles ABD and СBD are equal in all signs of equality of right-angled triangles. Q.E.D.