In the given figure, SP is the bisector of ∠RPT and PQRS is a cyclic quadrilateral. Prove that :

SQ = SR.

PQRS is a cyclic quadrilateral.

Sum of opposite angles in a cyclic quadrilateral = 180°.

∴ ∠QRS + ∠QPS = 180° ………..(1)

Also,

∠QPS + ∠SPT = 180° [As QPT is a straight line] ……..(2)

Subtracting equation (2) from (1) we get,

⇒ ∠QRS + ∠QPS - (∠QPS + ∠SPT) = 180° - 180°

⇒ ∠QRS - ∠SPT = 0

⇒ ∠QRS = ∠SPT ……….(3)

∠RQS = ∠RPS [Angles in same segment are equal] ……..(4)

∠RPS = ∠SPT [As PS bisects ∠RPT] …….(5)

From (3), (4) and (5) we get :

⇒ ∠QRS = ∠RQS.

∴ SQ = SR [As sides opposite to equal angles are also equal]

Hence, proved that SQ = SR.