1. **Stating the problem:** We need to find the values of angles $x$, $y$, and $z$ given some angles and relationships. 2. **Analyzing the given information:** The problem shows angles labeled $A$, $B$, $C$, $D$ with some known angles: $121^\circ$, $19^\circ$, and unknowns $x^\circ$, $y^\circ$, $z^\circ$. 3. **Assuming these angles form a straight line or triangle:** Since the problem does not specify, a common approach is to use the fact that angles on a straight line sum to $180^\circ$ or angles in a triangle sum to $180^\circ$. 4. **Using the straight line rule:** If $x$, $121^\circ$, and $z$ are on a straight line, then $$x + 121 + z = 180$$ 5. **Using the triangle sum rule:** If $x$, $y$, and $19^\circ$ form a triangle, then $$x + y + 19 = 180$$ 6. **Using the triangle sum rule for another triangle:** If $y$, $z$, and $A$ form a triangle, but since $A$ is unknown, we focus on the given data. 7. **From step 4:** $$x + z = 180 - 121 = 59$$ 8. **From step 5:** $$y = 180 - x - 19 = 161 - x$$ 9. **Assuming $y$ and $z$ are equal or related:** Since no other info is given, let's assume $y = z$. 10. **Substitute $z = y$ into step 7:** $$x + y = 59$$ 11. **From step 8:** $$y = 161 - x$$ 12. **Equate the two expressions for $y$:** $$161 - x = 59 - x$$ This is impossible unless $161 = 59$, so our assumption is wrong. 13. **Alternative approach:** Since the problem is incomplete, the best we can do is express $y$ and $z$ in terms of $x$: From step 7: $$z = 59 - x$$ From step 8: $$y = 161 - x$$ 14. **Final answers:** $$x = x$$ $$y = 161 - x$$ $$z = 59 - x$$ Without more information, these are the relationships between the angles.