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.
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