Question

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

1. its other two sides are equal.
2. its diagonals are equal.

Answer 1

Given:

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

To prove:

1. AD = BC
2. 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 ΔABC and ΔADB,

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]