If two sides of a cyclic quadrilateral are parallel, prove that:

(i) its other two sides are equal,

(ii) its diagonals are equal.

(i) Let ABCD be a cyclic quadrilateral in which AB || DC. AC and BD are its diagonals.

As, AB || DC (given)

∠DCA = ∠CAB [Alternate angles are equal]

Chord AD subtends ∠DCA and chord BC subtends ∠CAB at the circumference of the circle.

We know that,

Equal chords subtend equal angles at the circumference of a circle.

∴ chord AD = chord BC or AD = BC.

Hence, proved that AD = BC.

(ii) From figure,

⇒ ∠A + ∠C = 180° [As, sum of opposite angles in a cyclic quadrilateral = 180°]

Also,

⇒ ∠B + ∠C = 180° [Sum of co-interior angles = 180° (As, AB || CD)]

∴ ∠B + ∠C = ∠A + ∠C

⇒ ∠B = ∠A

In △ ACB and △ BDA,

⇒ AB = AB [Common side]

⇒ ∠B = ∠A [Proved above]

⇒ BC = AD [Proved above]

Hence, by SAS criterion of congruence.

△ ACB ≅ △ BDA

∴ AC = BD [By C.P.C.T.]

Hence, proved that diagonals are equal.