JAMB 1993 · UME · Q31

Circle geometry

In the figure, $PT$ is a tangent to the circle at $U$ and $QU \parallel RS$ . If $\angle TUR = 35^\circ$ and $\angle SRU = 50^\circ$ , find $x = \angle QRU$ .

Worked solution (try it first)

The angle between tangent $UT$ and chord $UR$ equals the angle in the alternate segment, so $\angle UQR = \angle TUR = 35^\circ$ .
$QU \parallel RS$ , so alternate angles are equal: $\angle QUR = \angle URS = 50^\circ$ .
The angles of triangle $QUR$ add up to $180^\circ$ : $x = 180^\circ - 35^\circ - 50^\circ$
$= 95^\circ$ , option A.

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