Mathematics
The given figure shows a circle with center O and ∠ABP = 42°. Calculate the measure of :
(i) ∠PQB
(ii) ∠QPB + ∠PBQ
Circles
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Answer
Join AP.
(i) We know that,
Angle in a semi-circle is a right angle.
∠APB = 90°.
In △APB,
⇒ ∠APB + ∠ABP + ∠BAP = 180° [Angle sum property of triangle]
⇒ 90° + 42° + ∠BAP = 180°
⇒ ∠BAP + 132° = 180°
⇒ ∠BAP = 180° - 132° = 48°.
From figure,
∠PQB = ∠BAP = 48° [Angles in same segment are equal]
Hence, ∠PQB = 48°.
(ii) In △BQP,
⇒ ∠QPB + ∠PBQ + ∠PQB = 180° [Angle sum property of triangle]
⇒ ∠QPB + ∠PBQ + 48° = 180°
⇒ ∠QPB + ∠PBQ = 180° - 48°
⇒ ∠QPB + ∠PBQ = 132°.
Hence, ∠QPB + ∠PBQ = 132°.
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