Question

Prove that non-isosceles trapeziums are not cyclic.

Answer 1

Consider the non-isosceles trapezium, ABCD.

Let us suppose that ABCD is a cyclic quadrilateral.

We know that opposite angles of a cyclic quadrilateral are supplementary.

⇒ ∠B + ∠D = 180°

⇒ ∠B = 180° − ∠D … (1)

ABCD is a non-isosceles trapezium and AB is parallel to CD

∴ ∠A + ∠D = 180° (Sum of the interior angles on the same side of the transversal is 180°.)

⇒ ∠A = 180° − ∠D … (2)

From equation (1) and equation (2), we get:

∠A = ∠B

This is not possible as ABCD is a non-isosceles trapezium.

Thus, our supposition was wrong.

Hence, non-isosceles trapeziums are not cyclic.