Mathematics
In the given figure, the two circles intersect at P and Q. If ∠A = 80° and ∠D = 84°, calculate :
(i) ∠QBC
(ii) ∠BCP
Circles
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Answer
PQAD is a cyclic quadrilateral as all vertices lie on the circumference of the circle.
Sum of opposite angles of cyclic quadrilateral = 180°
⇒ ∠DAQ + ∠DPQ = 180°
⇒ 80° + ∠DPQ = 180°
⇒ ∠DPQ = 180° - 80°
⇒ ∠DPQ = 100°.
Also,
⇒ ∠PDA + ∠PQA = 180°
⇒ 84° + ∠PQA = 180°
⇒ ∠PQA = 180° - 84°
⇒ ∠PQA = 96°.
Since exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
(i) ∠QBC = ∠DPQ = 100°.
Hence, ∠QBC = 100°.
(ii) ∠BCP = ∠PQA = 96°.
Hence, ∠BCP = 96°.
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