1. **State the problem:** We need to find the value of $x$, which is the measure of angle $\angle SWT$ in the given circle with points $S$ (center), $T$, $U$, $V$, and $W$ on the circumference. 2. **Given information:** - $\angle TWU = x^\circ$ - $\angle WUV = 35^\circ$ - $\angle UWV = 20^\circ$ - $\angle STU = 40^\circ$ 3. **Recall important rules:** - The sum of angles in triangle $WUV$ is $180^\circ$. - $S$ is the center, so $\angle STU$ is a central angle. - Angles subtended by the same chord in a circle are equal. 4. **Find $\angle WUV$ and $\angle UWV$ sum:** $$35^\circ + 20^\circ = 55^\circ$$ 5. **Calculate $\angle TWU = x$ using triangle $WUV$:** $$x + 35^\circ + 20^\circ = 180^\circ$$ $$x + 55^\circ = 180^\circ$$ $$x = 180^\circ - 55^\circ = 125^\circ$$ 6. **Interpret $\angle SWT$:** Since $S$ is the center and $W$ and $T$ are points on the circle, $\angle SWT$ is a central angle subtending arc $WT$. 7. **Use the fact that $\angle STU = 40^\circ$ is a central angle subtending arc $TU$. Since $\angle TWU = x = 125^\circ$ is an inscribed angle subtending arc $TU$, the central angle is twice the inscribed angle subtending the same arc.** 8. **Check consistency:** $$\angle STU = 2 \times \angle TWU$$ $$40^\circ = 2 \times 125^\circ$$ This is false, so $\angle TWU$ is not subtending the same arc as $\angle STU$. 9. **Since $\angle TWU = x$ is an inscribed angle, and $\angle SWT$ is a central angle subtending the same arc $TU$, then:** $$\angle SWT = 2 \times x$$ 10. **Calculate $\angle SWT$:** $$\angle SWT = 2 \times 125^\circ = 250^\circ$$ **Final answer:** $$\boxed{250^\circ}$$