JAMB 1991 · UME · Q36

Circle geometry

In the figure, $PMN$ and $PQR$ are two secants of the circle $MQTRN$ and $PT$ is a tangent. If $\angle PNR = 110^\circ$ and $\angle PMQ = 55^\circ$ , find $\angle MPQ$ .

Not to scale: the same figure serves both questions.

Worked solution (try it first)

$MQRN$ is a cyclic quadrilateral.
An exterior angle of a cyclic quadrilateral equals the interior angle opposite it, so $\angle QRN = \angle PMQ = 55^\circ$ .
$P$ , $Q$ and $R$ are on a straight line, so in triangle $PNR$ the angle at $R$ is $55^\circ$ and the angle at $N$ is $110^\circ$ .
The angles add up to $180^\circ$ : $\angle MPQ = 180^\circ - 110^\circ - 55^\circ$
$= 15^\circ$ , option D.

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