WAEC Mathematics 2014 Paper

Concept:Use the tangent–chord theorem and the property of parallel lines to find the unknown angle in $\triangle URT$.Explanation:Given: $PQ$ is a tangent to the circle at $R$, and $UT \parallel PQ$.Also, $\angle TRQ = x^\circ$.By the tangent–chord theorem, the angle between tangent $RQ$ and chord $RT$ equals the angle subtended by $RT$ in the alternate segment.Therefore, $\angle RUT = \angle TRQ = x^\circ$.Since $UT \parallel PQ$, and $RT$ is a transversal, the alternate interior angles are equal.Thus, $\angle UTR = \angle TRQ = x^\circ$.Now consider $\triangle URT$.The sum of interior angles of a triangle is $180^\circ$.So, $\angle RUT + \angle UTR + \angle URT = 180^\circ$.Substitute the known angles:$x^\circ + x^\circ + \angle URT = 180^\circ$.Simplify: $2x^\circ + \angle URT = 180^\circ$.Therefore, $\angle URT = 180^\circ - 2x^\circ$.Answer:$\angle URT = (180 - 2x)^\circ$ This corresponds to option D.