1. **State the problem:** We need to find the measures of the missing angles $x$ and $y$ given that one angle is $34^\circ$, $x = 146^\circ$, and $y = 90^\circ$. 2. **Analyze the given information:** The problem states $x = 146^\circ$ and $y = 90^\circ$. Since these are given, we just verify the relationships. 3. **Check angle relationships:** The angles around a point sum to $360^\circ$. If $x = 146^\circ$ and $y = 90^\circ$, and there is an angle of $34^\circ$, then the remaining angle should satisfy: $$146^\circ + 90^\circ + 34^\circ + \text{missing angle} = 360^\circ$$ 4. **Calculate the missing angle:** $$\text{missing angle} = 360^\circ - (146^\circ + 90^\circ + 34^\circ)$$ $$= 360^\circ - 270^\circ = 90^\circ$$ 5. **Conclusion:** The missing angle is $90^\circ$. The given $x$ and $y$ are already provided, so the problem is consistent. **Final answers:** $$x = 146^\circ$$ $$y = 90^\circ$$ $$\text{missing angle} = 90^\circ$$