Question

If a pair of opposite sides of a cylic quadrilateral are equal, prove that its diagonals are equal. [3 MARKS]

Answer 1

Concept: 1 Mark
Steps: 1 Mark
Answer: 1 Mark

Given: A cyclic quadrilateral ABCD in which AB = DC.

To Prove: Diagonal AC = Diagonal BD.

Proof :

$\angle BAC=\angle BDC$ ..... (i) [Angles in the same segment of a circle. ]

$\angle CAD=\angle ADB$ .....(ii) [ Equal arcs CD and AB subtend equal angles at the circumference ]

$\angle BAC+\angle CAD=\angle BDC+\angle ADB$ [Adding (i) and (ii)]

$\Rightarrow \angle BAD=\angle ADC$

$\Rightarrow BD=AC$ [Equal angles on the same circle cut off equal chords.]

$\Rightarrow diagonalAC=diagonalBD.$

Hence Proved.