1. The problem is to 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. 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. Here, we assume these angles form a straight line or a polygon where the sum of angles is $360^\circ$. 3. Sum all given angles: $$44^\circ + 52^\circ + y^\circ + x^\circ + 97^\circ = 360^\circ$$ 4. Combine known angles: $$44 + 52 + 97 = 193$$ 5. Substitute and simplify: $$193 + x + y = 360$$ 6. Isolate $x + y$: $$x + y = 360 - 193$$ $$x + y = 167$$ 7. Without additional information, we cannot find unique values for $x$ and $y$, but if $x$ and $y$ are complementary or supplementary, or if another relation is given, we can solve further. Since no other relation is provided, the solution is: $$x + y = 167^\circ$$ Final answer: $x + y = 167^\circ$