Answer
$\angle$ROS = 105$^{\circ}$
$\angle$POQ = 35$^{\circ}$
$\angle$ROQ = 40$^{\circ}$
$\angle$TOS = 35$^{\circ}$
Work Step by Step
$\angle$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$^{\circ}$
