SOLUTION: Prove that if the diagonals of a trapezoid are congruent, then the trapezoid is isosceles. How do I solve this proof? I know that I can start with trapezoid ABCD (given), but I'm

Question 846102: Prove that if the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.
How do I solve this proof? I know that I can start with trapezoid ABCD (given), but I'm unsure of how to go on. I appreciate any help
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Suppose that in trapezoid

and

. Denote by E the intersection of AC and BD.

Because triangles ABE and CDE are similar by angle-angle-angle, we can write DE = x, CE = y, then AE = ky and BE = kx. Since AC = BD, we have ky + y = kx + x --> y(k+1) = x(k+1), which implies x = y. Therefore DE = CE and AE = BE.

It follows that triangles ABE and CDE are isosceles, so

, and by SAS, triangles ABC and BAD are congruent, so AD = BC. The trapezoid is isosceles.
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