#### Question

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

(i) its other two sides are equal.

(ii) its diagonals are equal.

#### Solution

Given –

ABCD is a cyclic quadrilateral in which AB ∥ DC. AC and BD are its diagonals.

To prove –

(i) AD = BC

(ii) AC = BD

Proof –

(i) AB || DC ⇒ ∠DCA = ∠CAB [Alternate angles]

Now, chord AD subtends ∠DCA and chord BC subtends ∠CAB

At the circumference of the circle.

∴ ∠DCA = ∠CAB [proved]

∴ Chord AD = Chord BC or AD = BC

(ii) Now in ∠ABC and ∠ADB ,

AB = AB [Common]

∠ACB = ∠ADB [Angles in the same segment]

BC = AD [Proved]

By Side – Angle – Side criterion of congruence, we have

ΔACB ≅ ΔADB [SAS postulate]

The corresponding parts of the congruent triangles are congruent.

∴ AC = BD [c.p.c.t]