1. **State the problem:** We need to find the measures of the missing angles $x$ and $y$ given that $x=146^\circ$ and $y=90^\circ$ and one angle is $34^\circ$. 2. **Analyze the given information:** The angles $x$ and $y$ are related to the given angles by the properties of intersecting lines and adjacent angles. 3. **Use the linear pair rule:** Adjacent angles on a straight line sum to $180^\circ$. 4. **Calculate the missing angle adjacent to $34^\circ$:** $$x = 180^\circ - 34^\circ = 146^\circ$$ This matches the given $x$. 5. **Calculate the angle opposite to the right angle $90^\circ$:** Opposite angles formed by intersecting lines are equal, so $$y = 90^\circ$$ which matches the given $y$. 6. **Conclusion:** The missing angles are $x=146^\circ$ and $y=90^\circ$ as given.