Consider a trapezium ABCD with AB | |CD and BC = AD.
Draw AM ∠ CD and BN ∠ CD.
In ΔAMD and ΔBNC,
AD = BC (Given)
∠AMD = ∠BNC (By construction, each is 90°)
AM = BM (Perpendicular distance between two parallel lines is same)
ΔAMD ≅ ΔBNC (RHS congruence rule)
∠ADC = ∠BCD (CPCT) ... (1)
∠BAD and ∠ADC are on the same side of transversal AD.
∠BAD + ∠ADC = 180° ... (2)
∠BAD + ∠BCD = 180° [Using equation (1)]
This equation shows that the opposite angles are supplementary.
Therefore, ABCD is a cyclic quadrilateral.
