1. **Problem Statement:** Find the values of angles $x$ and $y$ given the angles $44^\circ$, $52^\circ$, $y^\circ$, $x^\circ$, and $97^\circ$ in a geometric figure. 2. **Key Concept:** The sum of angles around a point or in a triangle is $180^\circ$ or $360^\circ$ depending on the figure. We use the rule that the sum of angles on a straight line is $180^\circ$. 3. **Step 1:** Identify relationships. If $44^\circ$ and $52^\circ$ are adjacent to $y^\circ$ and $x^\circ$ respectively, and $97^\circ$ is given, we can set up equations based on angle sums. 4. **Step 2:** Assume $x$ and $y$ are angles adjacent to $44^\circ$ and $52^\circ$ such that: $$x + 44^\circ = 97^\circ$$ $$y + 52^\circ = 97^\circ$$ 5. **Step 3:** Solve for $x$: $$x = 97^\circ - 44^\circ = 53^\circ$$ 6. **Step 4:** Solve for $y$: $$y = 97^\circ - 52^\circ = 45^\circ$$ 7. **Final Answer:** $$x = 53^\circ$$ $$y = 45^\circ$$