1. **Stating the problem:** We are given a circle with center O and a central angle $\alpha = 288^\circ$. Points A, B, C, D, and H lie on the circumference, and angles $x$, $y$, and $z$ are marked at points C and A. We need to find the values of these angles based on the given information. 2. **Key formulas and rules:** - The central angle $\alpha$ subtends an arc on the circle. - The measure of an inscribed angle is half the measure of the intercepted arc. - The sum of angles around a point is $360^\circ$. 3. **Find the arc measure subtended by $\alpha$:** Since $\alpha = 288^\circ$ is a central angle, it intercepts an arc of $288^\circ$. 4. **Calculate the remaining arc:** The full circle is $360^\circ$, so the other arc intercepted by the points is $360^\circ - 288^\circ = 72^\circ$. 5. **Find angle $x$ at point C:** Angle $x$ is an inscribed angle subtending the arc opposite to $\alpha$, so $$x = \frac{1}{2} \times 288^\circ = 144^\circ.$$ 6. **Find angle $y$ at point C:** Angle $y$ subtends the smaller arc of $72^\circ$, so $$y = \frac{1}{2} \times 72^\circ = 36^\circ.$$ 7. **Find angle $z$ at point A:** Angle $z$ is an inscribed angle subtending the same arc as angle $y$, so $$z = y = 36^\circ.$$ **Final answers:** $$x = 144^\circ, \quad y = 36^\circ, \quad z = 36^\circ.$$