1. **State the problem:** Given that $\angle SUT \cong \angle YXZ$, prove that the vectors $\overrightarrow{WY}$ and $\overrightarrow{TV}$ are parallel. 2. **Analyze the given information:** The angles $\angle SUT$ and $\angle YXZ$ are congruent. These angles are formed by the intersection of the lines containing the vectors $\overrightarrow{WY}$ and $\overrightarrow{TV}$. 3. **Identify the relationship between the vectors:** Since the angles formed by these vectors are congruent, the direction of the vectors must be related. 4. **Use the property of parallel vectors:** Two vectors are parallel if their direction vectors are scalar multiples of each other, or equivalently, if the angles between them are equal. 5. **Conclude:** Because $\angle SUT \cong \angle YXZ$, the vectors $\overrightarrow{WY}$ and $\overrightarrow{TV}$ have the same direction angle, so they are parallel. **Final answer:** $\overrightarrow{WY} \parallel \overrightarrow{TV}$.