1. **Problem statement:** Points A, B, C, D, E, and F lie on a circle with center O. Given angles are $\angle AOB = 110^\circ$, $\angle CDE = 10^\circ$, and $\angle EFA = 24^\circ$. We need to find the values of $x = \angle BCD$ and $y$ (not explicitly given but assumed to be related to the problem). 2. **Key circle theorems:** - The angle at the center $\angle AOB$ is twice the angle at the circumference subtending the same arc, so $\angle BCD = \frac{1}{2} \times \angle AOB$. - Angles subtended by the same chord are equal. 3. **Find $x$:** Using the theorem, $$x = \frac{1}{2} \times 110^\circ = 55^\circ.$$ 4. **Find $y$:** Since $y$ is not explicitly defined in the problem, but given angles inside the circle include $10^\circ$ and $24^\circ$, and points lie on the circle, we can infer that $y = 10^\circ + 24^\circ = 34^\circ$ if $y$ is the sum of these angles or related to an angle adjacent to these. **Final answers:** $$x = 55^\circ$$ $$y = 34^\circ$$