JAMB 1989 · UME · Q31

Circle geometry

$MN$ is a tangent to the circle at $M$ , and $MR$ and $MQ$ are two chords, with $R$ , $Q$ and $N$ on a straight line. If $\angle QMN = 60^\circ$ and $\angle MNQ = 40^\circ$ , find $\angle RMQ$ .

Worked solution (try it first)

Tangent–chord: the angle between tangent $MN$ and chord $MQ$ equals the angle in the alternate segment, so $\angle MRQ = \angle QMN = 60^\circ$ .
Triangle $MQN$ : $\angle MQN = 180^\circ - 60^\circ - 40^\circ$
$= 80^\circ$ .
Angles on the straight line $RQN$ give $\angle MQR = 180^\circ - 80^\circ$
$= 100^\circ$ .
Triangle $RMQ$ : $\angle RMQ = 180^\circ - 60^\circ - 100^\circ$
$= 20^\circ$ , option D.

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