Question

A pair of opposite sides of a cyclic quadrilateral is equal. Prove that the other pair is parallel and its diagonals are also equal.

Answer 1

the opposite angle of a cyclic quadrilateral are supplementary,prove that the diagonal is a diameter so their are equal.

Answer 2

A cyclic quadrilateral ABCD, in which AD = BC. 

ad
b. c
To prove: AC = BD

Proof:

We know that equal chords subtend equal chords at the center of circle.

and also, the angle subtended by a chord at the center is twice the angle subtended by it at remaining part of the circle.

So, ∠AOD = ∠BOC        (O is the center of the circle)

And ∠AOD = 2∠ACD     (angle subtended by chord AD)

and ∠BOC = 2∠BDC       (angle subtended by chord BC)

Since, ∠AOD = ∠BOC,

∠ACD = ∠BDC          (1)

∠ACB = ∠ADB          (2)    (angles in the same segment are equal)

Adding (1) and (2),

∠BCD = ∠ADC          (3)

In triangles, ΔACD and ΔBDC,

CD = CD          (common)

∠BCD = ∠ADC      (using (3))

AD = BC            (given)

Hence,  ΔACDΔBDC  (SAS congruency rule)

AC = BD           (cpct)

Hence proved.