PQRS is a cyclic quadrilateral. Given ∠QPS = 73°, ∠PQS = 55° and ∠PSR = 82°, calculate ∠QRS, ∠RQS and ∠PRQ.

We know that,

In a cyclic quadrilateral, the sum of opposite angles is 180°.

⇒ ∠QPS + ∠QRS = 180°

⇒ 73° + ∠QRS = 180°

⇒ ∠QRS = 180° - 73°

⇒ ∠QRS = 107°.

From figure,

⇒ ∠PSR + ∠PQR = 180°

⇒ ∠PSR + ∠PQS + ∠RQS = 180°

⇒ 82° + 55° + ∠RQS = 180°

⇒ 137° + ∠RQS = 180°

⇒ ∠RQS = 180° - 137°

⇒ ∠RQS = 43°.

By angle sum property of a triangle we get,

⇒ ∠PSQ + ∠PQS + ∠QPS = 180°

⇒ ∠PSQ + 55° + 73° = 180°

⇒ ∠PSQ + 128° = 180°

⇒ ∠PSQ = 180° - 128°

⇒ ∠PSQ = 52°.

∠PSQ = ∠PRQ = 52° [Angles in the same segment]

Hence, ∠QRS = 107°, ∠RQS = 43° and ∠PRQ = 52°.