1. **Problem Statement:** Calculate the size of angle $x$ at point $A$ on the circumference of a circle with center $O$, given that angle $D$ on the circumference is $142^\circ$. 2. **Relevant Theorem:** The angle subtended by the same chord at the circumference is equal. Also, the angle at the center is twice the angle at the circumference subtended by the same chord. 3. **Step-by-step solution:** - Given angle $D = 142^\circ$ is an angle at the circumference subtended by chord $CE$. - Since $x$ is the angle subtended by the same chord $CE$ at point $A$ on the circumference, by the **Angles in the Same Segment Theorem**, angles subtended by the same chord in the same segment are equal. - Therefore, $x = 142^\circ$. 4. **Reasoning:** This follows from the property of circle geometry that angles subtended by the same chord at the circumference are equal. **Final answer:** $$x = 142^\circ$$ This completes the solution with the justification based on the circle theorem about angles subtended by the same chord.