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