1. **State the problem:** We need to find the values of $x$ and $y$ in the given circle diagrams and the length of line segment $CE$. 2. **Analyze the first diagram:** The angle at point $C$ is $100^\circ$. Since $A$, $B$, $C$, and $D$ lie on the circle, the opposite angles in the cyclic quadrilateral sum to $180^\circ$. 3. **Use the cyclic quadrilateral property:** $$\angle C + \angle A = 180^\circ$$ Given $\angle C = 100^\circ$, then $$\angle A = 180^\circ - 100^\circ = 80^\circ$$ 4. **Find $x$:** The angle $x$ is at point $B$, which subtends the same arc as $\angle A$. Therefore, $$x = \angle A = 80^\circ$$ 5. **Analyze the second diagram:** Points $C$, $E$, and $D$ are collinear with $E$ between $C$ and $D$. Given $CD = 8$ cm and $CA = 5$ cm. 6. **Use the right triangle formed by $E$, $F$, and $C$:** Since $EF$ is perpendicular to $CD$ at $E$, triangle $CEF$ is right angled at $E$. 7. **Find $CE$:** Since $C$, $A$, and $E$ lie on the circle and $CA = 5$ cm, and $CD = 8$ cm, with $E$ between $C$ and $D$, then $$CE + ED = CD = 8$$ 8. **Use the Pythagorean theorem in triangle $CEF$ if needed:** However, without additional information about $F$ or $y$, we cannot find $CE$ directly. 9. **Assuming $y$ is the length $EF$ and $F$ lies on the circle, use the power of a point or other circle properties to find $CE$ and $y$ if more data is given.** Since the problem does not provide enough information to find $y$ or $CE$ explicitly, the only value we can determine confidently is: $$x = 80^\circ$$