JAMB 1998 · UME · Q28

$TQ$ is a tangent to the circle $XYTR$ . If $\angle YXT = 32^\circ$ and $\angle RTQ = 40^\circ$ , find $\angle YTR$ .

Worked solution (try it first)

The angle between tangent $TQ$ and chord $TR$ equals the angle in the alternate segment, so $\angle TXR = \angle RTQ = 40^\circ$ .
So the whole angle at $X$ is $\angle YXR = 32^\circ + 40^\circ$
$= 72^\circ$ .
Opposite angles of cyclic quadrilateral $XYTR$ add up to $180^\circ$ : $\angle YTR = 180^\circ - 72^\circ$
$= 108^\circ$ , option A.