Question : PQRS is a cyclic quadrilateral and PQ is the diameter of the circle. If $\angle RPQ=38°$, what is the value (in degree) of $\angle PSR$?
Option 1: 52°
Option 2: 77°
Option 3: 128°
Option 4: 142°
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Correct Answer: 128°
Solution :
Given: PQRS is cyclic quadrilateral and PQ is the diameter of the circle and $\angle RPQ=38°$.
We know that the sum of the opposite angles of a cyclic quadrilateral is 180°. Also, the sum of all the angles in the triangle is 180°. Also, at any point on the circle, the diameter subtends a 90° angle.
Since PQ is the diameter of the circle. So, $\angle QRP=90°$.(Angle subtended by the diameter)
In $\triangle PRQ$,
$\angle RPQ+\angle PQR+\angle QRP=180°$
⇒ $\angle PQR=180°-90°-38°$
⇒ $\angle PQR=52°$
Since PQRS is a cyclic quadrilateral.
So, $\angle PQR+\angle PSR=180°$
⇒ $\angle PSR=180°-52°$
⇒ $\angle PSR=128°$
Hence, the correct answer is 128°.
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