1. **Stating the problem:** We are given a circle with points A, B, C, and D on the circumference. Inside the circle, triangle BAD is formed. The angle at point B is $35^\circ$, the central angle between points B and C is $110^\circ$, and we need to find the angle $x$ at point D. 2. **Relevant theorem:** The angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any point on the circumference. This is known as the **Angle at the Center Theorem**. 3. **Applying the theorem:** The central angle $110^\circ$ corresponds to the arc BC. The angle at point D, $x$, subtends the same arc BC on the circumference. 4. **Formula:** $$x = \frac{1}{2} \times 110^\circ = 55^\circ$$ 5. **Conclusion:** The angle $x$ at point D is $55^\circ$.