If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Given: In Trapezium ABCD, AB ∥ DC and AD = BC.
Construction: Draw AD ∥ BE.
Proof: In Quadrilateral ABCD,
AB ∥ DE [∵ Given]
AD ∥ BE [∵ By construction]
Hence, ABED is a Parallelogram.
AD = BE [∵ opposite sides of a parallelogram are equal]
AD = BC [∵ Given]
⇒ BE = BC
In ΔEBC, BE = BC [∵ Proved above]
Hence, ∠C = ∠2 …(1) [∵ In an isosceles triangle, the angles opposite to equal sides are equal]
∠A = ∠1 …(2) [∵ Opposite angles of a parallelogram are equal]
Here, ∠1 + ∠2 = 180° [∵ Linear pair]
⇒ ∠A + ∠C = 180° [∵ From the equation (1) and (2)]
⇒ ABED is a cyclic quadrilateral.