In an isosceles triangle ABC with base AC, the bisector AD cuts off a triangle CAD similar to ABC.

| January 30, 2021 | education

In an isosceles triangle ABC with base AC, the bisector AD cuts off a triangle CAD similar to ABC. Find the angles of triangle ABC.

Since, by condition, the triangles ABC and ACD are similar, and the triangles have a common angle, then in the ACD triangle the angle ADC = ACD, and therefore the ACD triangle is isosceles, AC = AD, and the triangles are similar in two angles.

Let the magnitude of the angle ACD = 2 * X0, then the angle of the CAD = 2 * X / 2 = X0, since the AD is the bisector of angle A.

Then in the ACD triangle the sum of the internal angles is: (X + 2 * X + 2 * X) = 180.

5 * X = 180.

X = 180/5 = 36.

Then the angle ACD = ASB = BAC = 2 * 36 = 72.

Then the angle ABC = (180 – 72 – 72) = 36.

Answer: The angles of triangle ABC are 36, 72, 72.



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