WAEC Mathematics 2015 Paper

Concept:The sum of the interior angles of a pentagon is fixed, so we use it to find the unknown angles from the given ratio.Explanation:A pentagon has $5$ interior angles. The sum of its interior angles is $(5-2)\times 180^\circ = 540^\circ$. Let the two angles in the ratio $2:3$ be $2x^\circ$ and $3x^\circ$. The other three angles are each $60^\circ$, so their total contribution is $3 \times 60^\circ = 180^\circ$.Now add all five angles and set the total equal to $540^\circ$:$$2x^\circ + 3x^\circ + 180^\circ = 540^\circ$$$$5x^\circ + 180^\circ = 540^\circ$$Subtract $180^\circ$ from both sides:$$5x^\circ = 360^\circ$$Divide by $5$:$$x^\circ = 72^\circ$$Thus the two angles are:$$2x = 2 \times 72^\circ = 144^\circ \quad \text{and} \quad 3x = 3 \times 72^\circ = 216^\circ$$Since the ratio is $2:3$, the smaller of the two angles is $2x$.Therefore, the smaller of the two angles is $144^\circ$.Answer:D. $144^\circ$