Question

If a pair of opposite sides of a cyclic quadrilateral are equal then prove that its diagonals are equal.

Answer 1

Let ABCD be a cyclic quadrilateral and AB = CD.
⇒ arc AB = arc CD (Corresponding arcs of the equal chords) Adding arc AD to both the sides of the equation; arc AB + arc AD = arc CD + arc AD ∴ arc BAD = arc CDA
⇒ Chord BD = Chord CA
⇒ BD = CA Hence, when pair of opposite sides of a cyclic quadrilateral are equal, diagonals are also equal.
hope its help dear!!

Answer 2

Answer:

Let the quadrilateral be ABCD with AD = BC.

Since AD = BC, ABCD is an isosceles trapezium

Angle D = angle C - - - - - - (1)

Join AC.

In triangle ADC and BCD

CD = CD (common)

angle D = angle C (from 1)

AD = BC

Triangle ADC is congruent to BCD by SAS criterion

AC = BD by CPCT

Hence proved