The angle at the apex of an isosceles triangle is 36 degrees. Prove that the bisector of an angle at the base

The angle at the apex of an isosceles triangle is 36 degrees. Prove that the bisector of an angle at the base divides a given triangle into two isosceles triangles.

According to the condition, the triangle ABC is isosceles, then the angles at the base of the AC are equal.

Then the angle BAC = BCA = (180 – ABC) / 2 = (180 – 36) / 2 = 144/2 = 72.

The segment AH is the bisector of the angle BAC, then the angle BAH = CAH = BAC / 2 = 72/2 = 36.

In triangle ABH, angle ABH = BAH = 36, then triangle ABH is isosceles, AH = BH.

In the ACH triangle, the angle AHC = (180 – ACH – CAH) = (180 – 72 – 36) = 72.

Then the angle ACH = AHC = 72, then the triangle ACH is isosceles, AH = AC, which was required to prove.