The diagonal of a trapezoid lies on the bisector of its corresponding angles.

The diagonal of a trapezoid lies on the bisector of its corresponding angles. Prove that the two sides of this trapezoid are equal. Can this trapezoid be called isosceles?

By condition, AC is the bisector of the BAD angle, therefore, the BAC angle will be equal to the CAD angle.

Consider a triangle ABC, in which the angle BCA is equal to the angle CD of the triangle ACD as criss-crossing angles at the intersection of parallel lines BC and AD secant AC.

Then the angle BAC = ACB, and therefore, the triangle ABC will be isosceles, since the angles at the base are equal. Side AB is equal to the base of BC. Q.E.D.

The trapezoid cannot be called isosceles under these conditions, since the BCD angle can take different values.