1. **Problem statement:** We are given a circle with center O and points A, B, C, D, and E on the circumference. We need to find the size of angle $x$ at point A inside the circle, given that angle $148^\circ$ is marked at point D on the circumference. 2. **Key fact:** 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. 3. Since $\angle D = 148^\circ$ is an inscribed angle, the central angle subtending the same arc is: $$\text{Central angle} = 2 \times 148^\circ = 296^\circ$$ 4. The full circle is $360^\circ$, so the reflex central angle corresponding to the other arc is: $$360^\circ - 296^\circ = 64^\circ$$ 5. Angle $x$ at point A subtends the same arc as the central angle of $64^\circ$, so: $$x = \frac{64^\circ}{2} = 32^\circ$$ 6. **Reason:** Angle $x$ is an inscribed angle subtending the arc corresponding to the $64^\circ$ central angle, so it is half of that central angle. **Final answer:** $$x = 32^\circ$$