Basic College Mathematics (9th Edition)

$\angle$ROS = 105$^{\circ}$ $\angle$POQ = 35$^{\circ}$ $\angle$ROQ = 40$^{\circ}$ $\angle$TOS = 35$^{\circ}$
$\angle$POU and $\angle$ROS are vertical angles. This means that they are equal. Therefore $\angle$ROS = 105$^{\circ}$ $\angle$UOT and $\angle$ROQ are vertical angles. This means that they are equal. Therefore $\angle$ROQ = 40$^{\circ}$ The sum of $\angle$POU, $\angle$POQ, and $\angle$UOT equals 180$^{\circ}$. This is due to $\angle$QOT being a Straight Angle. $\angle$POQ = 180$^{\circ}$ - (105$^{\circ}$ + 40$^{\circ}$) $\angle$POQ = 180$^{\circ}$ - 145$^{\circ}$ $\angle$POQ = 35$^{\circ}$ $\angle$POQ and $\angle$ROQ are vertical angles. This means that they are equal. Therefore $\angle$ROQ = 40$^{\circ}$