The sides of the trapezoid ABCD meet at point K. AB is the base of the trapezoid. triangle ABK is equilateral.

The sides of the trapezoid ABCD meet at point K. AB is the base of the trapezoid. triangle ABK is equilateral. Prove that the difference between the bases of a trapezoid is equal to its lateral edge

From the vertex C of the trapezoid, draw a straight line CM parallel to the lateral side AB.

The СDM triangle is similar to the AKD triangle in two angles, the angle D is common, the angle AKD and MCD are similar at the intersection of parallel straight lines AK and CM of the secant DK.

Then the triangle CDK is also equilateral, CM = CD = DK.

Quadrangle ABCM is a parallelogram, then AM = BC. AD = AM + DM = BC + CD.

Then СD = АD – ВС, as required.