JAMB 1983 · UME · Q36

Circle geometry

$PQRS$ is a cyclic quadrilateral in which $PQ = PS$ . $PT$ is a tangent to the circle and $PQ$ makes an angle of $50^\circ$ with the tangent, as shown. What is the size of $\angle QRS$ ?

Worked solution (try it first)

The angle between a tangent and a chord equals the angle in the alternate segment.
So $\angle PSQ = \angle QPT = 50^\circ$ .
$PQ = PS$ , so triangle $PQS$ is isosceles and $\angle PQS = \angle PSQ = 50^\circ$ .
Then $\angle QPS = 180^\circ - 50^\circ - 50^\circ$
$= 80^\circ$ .
Opposite angles of a cyclic quadrilateral add up to $180^\circ$ , so $\angle QRS = 180^\circ - 80^\circ$
$= 100^\circ$ , option E.

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