JAMB 1988 · UME · Q38

In the figure, $XR$ and $YQ$ are tangents to the circle $YZXP$ . If $\angle ZXR = 45^\circ$ and $\angle YZX = 55^\circ$ , find $\angle ZYQ$ .

Worked solution (try it first)

Tangent–chord at $X$ : the angle between tangent $XR$ and chord $XZ$ equals the angle in the alternate segment, so $\angle ZYX = 45^\circ$ .
The angles of triangle $XYZ$ add up to $180^\circ$ : $\angle YXZ = 180^\circ - 55^\circ - 45^\circ$
$= 80^\circ$ .
Tangent–chord at $Y$ : the tangent makes $80^\circ$ with chord $YZ$ on the side away from $Q$ .
So $\angle ZYQ = 180^\circ - 80^\circ$
$= 100^\circ$ , option C.