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

| April 28, 2021 | education

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

Let us prove that triangles ABD = ACD.

Since the trapezoid ABCD is isosceles, then AB = CD. Triangles ABD and ACD have a common side AD. The angle BAD of the triangle ABD is equal to the angle CDA of the triangle ADC.

Therefore, triangles ABD and ACD are similar in two sides and the angle between them. The coefficient of their similarity is 1, therefore, ABD = ACD.

Let us prove that triangles ABO = CDO.

By the property of an isosceles trapezoid, its diagonals, at the point of intersection, are divided into correspondingly equal segments. Then ВO = CO and AO = DO.

The sides AB and CD of triangles ABO and COD are equal as the lateral sides of an isosceles trapezoid.

Then the triangles ABO and COD are equal on three sides.

Q.E.D.



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