1. **Problem:** Find the values of $x$, $y$, and $z$ in the given circles with center $O$ where angles around the center sum to $360^\circ$. 2. **Formula:** The sum of angles around a point is always $360^\circ$. 3. **Step-by-step solutions:** (a) Angles: $x$, $70^\circ$, $30^\circ$, $y$, $\alpha$, $\beta$. $$x + 70 + 30 + y + \alpha + \beta = 360$$ Without values for $\alpha$ and $\beta$, $x$ and $y$ cannot be uniquely determined. (b) Angles: $150^\circ$, $\gamma$, $x$, $\theta$, $\alpha$, $y$, $\beta$. $$150 + \gamma + x + \theta + \alpha + y + \beta = 360$$ Again, without values for other variables, $x$ and $y$ cannot be found. (c) Angles inside circle: $x$, $240^\circ$, $55^\circ$, $\theta$, $\alpha$. Sum equals $360^\circ$: $$x + 240 + 55 + \theta + \alpha = 360$$ Simplify known angles: $$x + \theta + \alpha = 360 - 295 = 65$$ Without $\theta$ and $\alpha$, $x$ is undetermined. (d) Angles: $x$, $\beta$, $40^\circ$, $y$, $\alpha$. Sum: $$x + \beta + 40 + y + \alpha = 360$$ No unique solution without other values. (e) Angles: $50^\circ$, $60^\circ$, $x$, $\theta$, $\alpha$, $y$. Sum: $$50 + 60 + x + \theta + \alpha + y = 360$$ Simplify known angles: $$x + \theta + \alpha + y = 360 - 110 = 250$$ No unique solution without more info. (f) Circle inscribed in square ABCD with angles: $x$, $\alpha$, $y$, $110^\circ$, $\beta$, $z$. Sum: $$x + \alpha + y + 110 + \beta + z = 360$$ No unique solution without values for $\alpha$, $\beta$. **Final note:** To find exact values of $x$, $y$, and $z$, additional information or relationships between the variables are needed. --- **Summary:** The sum of angles around the center of a circle is $360^\circ$. Each problem sets up an equation summing the given angles to $360^\circ$. Without further data, the variables cannot be uniquely solved.