1. **Stating the problem:** We need to find the values of angles $x$ and $y$ given the angles $44^\circ$, $52^\circ$, and $97^\circ$ in a geometric figure involving points $A$, $B$, $C$, and $D$. 2. **Understanding the problem:** Typically, in such problems, the sum of angles around a point or in a triangle is used. The sum of angles in a triangle is $180^\circ$, and the sum of angles around a point is $360^\circ$. 3. **Using the sum of angles in a triangle:** Suppose $x$ and $y$ are angles in triangles formed with the given angles. We can write equations based on the sum of angles. 4. **Forming equations:** - For the triangle with angles $44^\circ$, $52^\circ$, and $y^\circ$: $$44 + 52 + y = 180$$ - Simplify: $$96 + y = 180$$ $$y = 180 - 96 = 84$$ 5. **For the triangle with angles $x^\circ$, $y^\circ$, and $97^\circ$:** $$x + y + 97 = 180$$ 6. **Substitute $y = 84$ into the equation:** $$x + 84 + 97 = 180$$ $$x + 181 = 180$$ 7. **Simplify:** $$x = 180 - 181 = -1$$ Since an angle cannot be negative, re-examine the problem. Possibly $x$ and $y$ are angles on a straight line or around a point. 8. **Using the straight line rule:** Angles on a straight line sum to $180^\circ$. - If $x$ and $97^\circ$ are supplementary: $$x + 97 = 180$$ $$x = 180 - 97 = 83$$ 9. **Using the sum of angles around point $B$ or $C$ to find $y$:** - If $y$, $44^\circ$, and $52^\circ$ are angles around a point, their sum is $360^\circ$: $$y + 44 + 52 = 360$$ $$y + 96 = 360$$ $$y = 360 - 96 = 264$$ This is too large for an angle in a triangle, so likely $y$ is supplementary to $44^\circ$ and $52^\circ$: $$y + 44 + 52 = 180$$ $$y + 96 = 180$$ $$y = 84$$ 10. **Final answers:** $$x = 83^\circ$$ $$y = 84^\circ$$