WAEC Mathematics 2021 Paper

Concept:The angle at the centre is twice the angle at the circumference, and a radius is equal to another radius, so isosceles triangle properties apply.Explanation:Since $\angle PQR = 72^\circ$, the angle at the centre subtending the same chord $PR$ is:$$\angle POR = 2 \times 72^\circ = 144^\circ.$$In $\triangle OPR$, $OP = OR$ because both are radii.Therefore, $\triangle OPR$ is isosceles and:$$\angle OPR = \angle ORP = \frac{180^\circ - 144^\circ}{2} = 18^\circ.$$Given that $OR \parallel PS$, and $RP$ acts as a transversal, alternate angles are equal:$$\angle RPS = \angle ORP = 18^\circ.$$From the diagram, $PR$ lies between $PO$ and $PS$, so:$$\angle OPS = \angle OPR + \angle RPS = 18^\circ + 18^\circ = 36^\circ.$$Answer:$$36^\circ$$Option D.