The bisector divides a triangle into two isosceles triangles. Find the angles of the original triangle?

Consider a triangle ABC, where the angle ABC is equal to α.

Let ВM be a bisector.

Then, the angles CBM = ABM = α / 2.

Since the side ВM cannot be equal to the sides AB and BC, then, according to the condition ВM = MС and ВM = MA.

Then the angle MAВ = angle ABM = α / 2, angle MСВ = angle CBM = α / 2.

MAВ + MСВ + ABC = 180º.

α / 2 + α / 2 + α = 180º

α = 90º.

Thus, triangle ABC is right-angled with right angle B and angles A and C equal to 45 degrees.

Answer: the angles of the original triangle are 90º, 45º, 45º.