WAEC Mathematics 2017 Paper

Concept:Use the tangent–chord theorem, the angle in a semicircle, and the fact that tangents from the same external point are equal.Explanation:Since $PQ$ and $PS$ are tangents from the same point $P$, we have $PQ = PS$.So $\triangle PQS$ is isosceles, and $\angle PSQ = \angle PQS = m$.Also, $\angle SPQ = n$, so$m + m + n = 180^\circ$$n = 180^\circ - 2m$.In $\triangle QRS$, $QR$ is a diameter, so $\angle QSR = 90^\circ$ because it is an angle in a semicircle.Given $\angle SQR = 33^\circ$,$\angle QRS = 180^\circ - 90^\circ - 33^\circ = 57^\circ$.By the tangent–chord theorem at $Q$,$\angle PQS = \angle QRS = 57^\circ$.Thus $m = 57^\circ$.Then$n = 180^\circ - 2(57^\circ) = 66^\circ$.Therefore,$m + n = 57^\circ + 66^\circ = 123^\circ$.Answer:$123^\circ$ (Option B).