(a)(i) < PTS = 38^\circ (alternate angle) 42^\circ + x + 38^\circ = 180^\circ (angles on a straight line) 80^\circ + x = 180^\circ x = 180^\circ - 80^\circ = 100^\circ (ii) < SQT = 42^\circ (alternate angle) < SQR = 60^\circ (sum of angles in a triangle) y + 42^\circ + 60^\circ = 180^\circ (angle on a straight line) y + 102^\circ = 180^\circ y = 180^\circ - 102^\circ = 78^\circ (b) \(< QSP = \frac{150}{2}\) (angle at the centre is twicw angle at the circumference) \(< QSP = 75^\circ\) \(\hat{R} + \hat{P} = 180^\circ\) (opposite angles in a cyclic quad are supplementary) \(\hat{P} + 75^\circ + 45^\circ = 180^\circ\) (sum of angle in a triangle) \(\hat{P} = 180^\circ - 120^\circ\) \(\hat{P} = 60^\circ\) \(\hat{R} + \hat{P} = 180^\circ\) \(\hat{R} = 180^\circ - 60^\circ\) \(\hat{R} = 120^\circ\) \(< RQS + < QRS + < QSR = 180^\circ\) (sum of angle in a triangle) \(< RQS = 180^\circ - (40^\circ + 120^\circ)\) = \(180^\circ - 160^\circ\) \(< RQS = 20^\circ\)