1. **State the problem:** We are given two expressions for line segments in a quadrilateral: $UV = x - 61$ and $TW = x - 63$. We need to find the value of $x$. 2. **Analyze the problem:** Since the problem involves segments $UV$ and $TW$ inside the quadrilateral and the figure shows marks indicating equal segments, it suggests that $UV$ and $TW$ might be equal. 3. **Set up the equation:** Assuming $UV = TW$, we write: $$x - 61 = x - 63$$ 4. **Solve the equation:** Subtract $x$ from both sides: $$x - 61 - x = x - 63 - x$$ which simplifies to: $$-61 = -63$$ 5. **Interpret the result:** The equation $-61 = -63$ is false, meaning our assumption that $UV = TW$ is incorrect or incomplete based on the given information. 6. **Re-examine the problem:** Since $UV$ and $TW$ are given as expressions involving $x$, and the figure shows marks indicating equal segments $SW = WV$ and $ST = TU$, but no direct equality between $UV$ and $TW$, we need more information or relationships to solve for $x$. 7. **Conclusion:** With the current information, the value of $x$ cannot be determined uniquely. **Final answer:** The value of $x$ cannot be found with the given information.