1. **Problem statement:** We are given a circle with center O and points A, B, C, D, and E on the circumference. We know the angle at point D is 148° and need to find the size of angle $x$ at point A. 2. **Key fact:** The angle subtended by the same chord at the center of the circle is twice the angle subtended at the circumference on the same side. 3. Since angle at D is 148°, and assuming angle $x$ at A is subtended by the same chord, the angle at the center O subtended by this chord is twice angle $x$. 4. The angle at the center O corresponding to the chord is given as 148° (angle at D is on the circumference, so angle at center is twice angle at circumference). But here, angle at D is 148°, so angle at center O subtended by the same chord is $2 \times 148^\circ = 296^\circ$ which is impossible since angles around a point sum to 360°. 5. Instead, angle at D (148°) and angle $x$ at A are angles subtended by the same chord but on opposite sides of the chord. The rule is that angles subtended by the same chord on opposite sides sum to 180°. 6. Therefore, angle $x + 148^\circ = 180^\circ$. 7. Solving for $x$: $$ x = 180^\circ - 148^\circ = 32^\circ $$ **Final answer:** $x = 32^\circ$. **Reason:** Angles subtended by the same chord on opposite sides of the chord sum to 180° (the supplementary angles property in a circle).