1. **Stating the problem:** We need to find the values of angles $x$, $y$, and $z$ given some angles and possibly relationships between them. 2. **Analyzing the given information:** The angles mentioned are $40^\circ$, $121^\circ$, and $0^\circ$. We also see $z^\circ$, $x^\circ$, and $y^\circ$ which are unknowns. 3. **Assuming these angles are part of a geometric figure (like a triangle or polygon), we use the angle sum rules:** - The sum of angles in a triangle is $180^\circ$. - The sum of angles around a point is $360^\circ$. 4. **Using the angle sum in a triangle:** Suppose $x$, $y$, and $40^\circ$ are angles of a triangle: $$x + y + 40 = 180$$ 5. **Using the angle sum in another triangle or polygon involving $z$ and $121^\circ$:** Suppose $z$, $121^\circ$, and $0^\circ$ are angles of another triangle or linear pair: $$z + 121 + 0 = 180$$ 6. **Solving for $z$:** $$z = 180 - 121 - 0 = 59$$ 7. **Solving for $x$ and $y$:** From step 4, $$x + y = 180 - 40 = 140$$ 8. **Without additional information, $x$ and $y$ can be any values such that $x + y = 140$ degrees.** **Final answers:** $$z = 59^\circ$$ $$x + y = 140^\circ$$ Since no further data is given, $x$ and $y$ cannot be uniquely determined.