In an isosceles triangle the vertex angle is thrice of either base angles. find the measures of each angle of the triangle?

Let the measure of each base angle be ( x ). Then, the vertex angle is ( 3x ). In an isosceles triangle, the sum of the angles is ( 180^\circ ), so we have the equation ( x + x + 3x = 180^\circ ). Simplifying gives ( 5x = 180^\circ ), leading to ( x = 36^\circ ). Thus, the measures of the angles are ( 36^\circ ) for each base angle and ( 108^\circ ) for the vertex angle.