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Question
PQRS is a cyclic quadrilateral. Given ∠QPS = 73°, ∠PQS = 55° and ∠PSR = 82°, calculate:
1) ∠QRS
2) ∠RQS
3) ∠PRQ
Solution
Given: PQRS is a cyclic quadrilateral.
∠QPS = 73°, ∠PQS = 55° and ∠PSR = 82°
1) Opposite angle of a cyclic quadrilateral is supplementary.
⇒ ∠QPS + ∠QRS = 180°
⇒ 73° + ∠QRS = 180°
⇒ ∠QRS = 180° - 73= 107°
2) Opposite angle of a cyclic quadrilateral are supplementary.
⇒ ∠PSR + ∠PQR = 180°
⇒ ∠PSR + (⇒ ∠PQS + ⇒ ∠RQS) = 180°
⇒ 82° + 55° + ∠RQS = 180°
⇒ ∠RQS = 180° - 137° = 43°
3) In ΔPQS, by angle sum property, we have
⇒ ∠PSQ + ∠PQS + ∠QPS = 180°
⇒ ∠PSQ + 55° + 73° = 180°
⇒ ∠PSQ = 180° - 128° = 52°
Now ⇒ ∠PRQ = ∠PSQ (angles in the same segment of a circle)
⇒ ∠PRQ = 52°
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