JAMB 1985 · UME · Q37

Circle geometry

In the figure, $MNQP$ is a cyclic quadrilateral; $MN$ and $PQ$ are produced to meet at $X$ , and $NQ$ and $MP$ are produced to meet at $Y$ . If $\angle MNQ = 86^\circ$ and $\angle NQP = 122^\circ$ , find $(x^\circ, y^\circ)$ , the angles at $X$ and $Y$ .

Worked solution (try it first)

Opposite angles of a cyclic quadrilateral add up to $180^\circ$ .
So $\angle NMP = 180^\circ - 122^\circ$
$= 58^\circ$ and $\angle MPQ = 180^\circ - 86^\circ$
$= 94^\circ$ .
$X$ is where $MN$ and $PQ$ meet, so use triangle $MXP$ : $x = 180^\circ - 58^\circ - 94^\circ$
$= 28^\circ$ .
$Y$ is where $NQ$ and $MP$ meet, so use triangle $MNY$ : $y = 180^\circ - 58^\circ - 86^\circ$
$= 36^\circ$ .
So $(x^\circ, y^\circ) = (28^\circ, 36^\circ)$ , option A.

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