JAMB 1987 · UME · Q35

Circle geometry

Two chords $QR$ and $NP$ of a circle intersect inside the circle at $X$ . If $\angle RQP = 37^\circ$ , $\angle RQN = 49^\circ$ and $\angle QPN = 35^\circ$ , find $\angle PRQ$ .

Worked solution (try it first)

Angles in the same segment are equal.
$\angle RPN$ and $\angle RQN$ both stand on arc $RN$ , so $\angle RPN = 49^\circ$ .
So $\angle QPR$ is $\angle QPN + \angle NPR$ , which is $35^\circ + 49^\circ = 84^\circ$ .
The angles of triangle $PQR$ add up to $180^\circ$ : $\angle PRQ = 180^\circ - 37^\circ - 84^\circ$
$= 59^\circ$ , option D.

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