1. The problem is to find the values of angles $x$, $y$, and $z$ given the angles $40^\circ$, $121^\circ$, and the variables $x^\circ$, $y^\circ$, $z^\circ$. 2. We use the rule that the sum of angles around a point or in a triangle is $180^\circ$ or $360^\circ$ depending on the figure. Since the problem does not specify the figure, we assume these angles form a straight line or triangle. 3. If these angles are on a straight line, their sum is $180^\circ$. So, $$x + y + z + 40 + 121 = 180$$ 4. Simplify the known angles: $$x + y + z + 161 = 180$$ 5. Subtract 161 from both sides: $$x + y + z = 180 - 161$$ $$x + y + z = 19$$ 6. Without additional information, we cannot find unique values for $x$, $y$, and $z$, but their sum is $19^\circ$. Final answer: $$x + y + z = 19^\circ$$