1. **Problem statement:** Find the missing angles $x$ and $y$ in the two circle diagrams. 2. **Key formulas and rules:** - The measure of an inscribed angle is half the measure of its intercepted arc. - The central angle is equal to the measure of its intercepted arc. - Angles subtending the same arc are equal. --- ### Diagram 1 (angle $x$): 3. Given: Angle at point $C$ inside the circle is $106^\circ$. 4. Since $C$ is on the circumference and $O$ is the center, angle $C$ is an inscribed angle subtending arc $AD$. 5. The central angle $COD$ (not shown) would be twice the inscribed angle $C$, so the arc $AD$ measures $2 \times 106^\circ = 212^\circ$. 6. Angle $x$ at point $B$ is an inscribed angle subtending arc $AC$. 7. The total circle is $360^\circ$, so arc $AC = 360^\circ - 212^\circ = 148^\circ$. 8. Therefore, $x = \frac{1}{2} \times 148^\circ = 74^\circ$. --- ### Diagram 2 (angle $y$): 9. Given: Angle at point $A$ subtending arc $BD$ is $52^\circ$. 10. Angle $A$ is an inscribed angle, so arc $BD = 2 \times 52^\circ = 104^\circ$. 11. Angle $y$ at point $C$ subtends the same arc $BD$ (since $C$ lies on the circumference and $y$ is the angle subtending arc $BD$). 12. Therefore, $y = 52^\circ$ (angles subtending the same arc are equal). --- **Final answers:** $$x = 74^\circ$$ $$y = 52^\circ$$