This problem involves a circle with intersecting lines, and we need to find the value of angle $x$.

Step-by-step solution:

1. Identify the key angles:

   • There is an inscribed angle of $35^\circ$, adjacent to the angle $x$.
   • An external angle $141^\circ$ is also given.

2. Recall the relationship between an exterior angle and the intercepted arc: The exterior angle theorem states that the exterior angle of a circle is equal to half the difference of the intercepted arcs.

   Specifically: $\text{Exterior Angle} = \frac{1}{2}(\text{Larger Intercepted Arc} - \text{Smaller Intercepted Arc})$ In this case, the angle $x$ is the exterior angle to the arc, so: $x = \frac{1}{2}(141^\circ - 35^\circ)$

3. Calculate $x$: $x = \frac{1}{2}(141^\circ - 35^\circ) = \frac{1}{2} \times 106^\circ = 53^\circ$

So, the value of $x$ is 53 degrees.

Would you like further clarification or details? Here are some related questions to expand on this:

1. What is the exterior angle theorem and how does it apply to circles?
2. How do we calculate the sum of opposite angles in a cyclic quadrilateral?
3. What are inscribed and central angles in a circle, and how do they relate?
4. How do intercepted arcs affect the calculation of angles in a circle?
5. Can the exterior angle of a triangle be applied similarly in circle problems?

Tip: When solving circle geometry problems, look for relationships between angles, arcs, and lines, as these often help simplify complex configurations.