Question

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

(i) ∠QRS 

(ii) ∠RQS 

(iii) ∠PRQ

Answer 1

(i) Since PQRS is a cyclic quadrilateral 

∠QPS + ∠QRS – 180° 

⇒ 73° + ∠QRS = 180° 

⇒ ∠QRS = 180° – 73° 

∠QRS = 107° 

(ii) Again, ∠PQR + ∠PSR = 180° 

∠PQS + ∠RQS + ∠PSR = 180° 

55° – ∠RQS + 82° = 180° 

∠RQS = 180° – 82° – 55° = 43° 

(iii) In ∆PQS, by using angles sum property of a ∆. 

∠PSQ + ∠SQP + ∠QPS = 180° 

∠PSQ + 55° + 73° = 180° 

∠PSQ = 180° – 55° – 73° 

∠PSQ = 52° 

Now, ∠PRQ = ∠PSQ = 52° [Oop. ∠s of the same segment] 

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