1. **Problem Statement:** Given that $\overrightarrow{SU}$ and $\overrightarrow{VX}$ are parallel lines and $m \angle UTR = 139^\circ$, find $m \angle VWY$. 2. **Understanding the problem:** The lines $SU$ and $VX$ are parallel vertical lines. The angle $UTR$ is formed at point $T$ on the diagonal line intersecting these vertical lines. We need to find the angle $VWY$ on the left vertical line. 3. **Key concept:** When a transversal crosses parallel lines, alternate interior angles are equal. Also, angles on a straight line sum to $180^\circ$. 4. **Step-by-step solution:** - Since $m \angle UTR = 139^\circ$, the angle adjacent to it on the straight line at $T$ is $180^\circ - 139^\circ = 41^\circ$. - Because $\overrightarrow{SU}$ and $\overrightarrow{VX}$ are parallel, the angle $UTR$ corresponds to an alternate interior angle with $m \angle WTY$ (angle at $T$ between $W$ and $Y$ on the left vertical line). - Therefore, $m \angle WTY = 139^\circ$. - The angle $VWY$ is vertically opposite to $W T Y$ (since $W$ is between $V$ and $Y$ on the vertical line), so $m \angle VWY = m \angle WTY = 139^\circ$. 5. **Final answer:** $$m \angle VWY = 139^\circ$$