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

Draw a trapezium PQRS with PQ || RS

PS and QR are the non-parallel sides that are equal. PS = QR. Draw PM ⊥ RS and QN ⊥ RS.

Consider ΔPMS and ΔQNR

PS = QR (Given)

∠PMS = ∠QNR (90°)

PM = QN (Perpendicular distance between two parallel lines is same)

By RHS congruence, ΔPMS ≅ ΔQNR.

Using CPCT, ∠PSR = ∠QRS.....(1)

∠QPS and ∠PSR are on the same side of transversal PS

∠QPS + ∠PSR = 180°

∠QPS + ∠QRS = 180° [From equation(1)]

Hence proved from above hat the sum of opposite angles is supplementary. Hence, PQRS is a cyclic quadrilateral.