In an isosceles trapezoid ABCD, the diagonals meet at point O. Prove that triangle ABD = ACD, ABO = CDO.

Let us prove the equality of triangles ABD and ACD.

In an isosceles trapezoid, the sides are equal, AB = CD of the angle at the base of the trapezoid are equal, the angle BAD = ADC, the angle ABC = BCD. For triangles, side AD is common, AB = CD and angle BAD = ADC, then triangles ABD and ACD are equal on two sides and the angle between them.

Let us prove the equality of triangles ABO and CDO.

By the property of the diagonals of an isosceles trapezoid, the diagonals, at the point of intersection, are divided into correspondingly equal segments, then OB = OС, OA = OD. Since the trapezoid is isosceles, then AB = CD, and then the triangles ABO and CDO are equal on three sides, which was required to prove