1. **Problem statement:** We have a circle with center $O$ and points $A$, $B$, $C$, and $D$ on the circumference. $FDE$ is a tangent line touching the circle at $D$. We are given two angles: $x^\circ$ at the center $O$ formed by $OB$ and $OD$, and $y^\circ$ at the circumference at point $A$ (angle $BAD$). We need to find the value of $y - x$. 2. **Relevant theorem:** The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment. This means the angle between tangent $FDE$ and chord $AD$ at $D$ equals the angle $BAD$ at the circumference. 3. **Key fact:** The angle at the center $O$ subtended by chord $BD$ is $x^\circ$. The angle at the circumference subtended by the same chord $BD$ is half of that, so it is $\frac{x}{2}^\circ$. 4. **Using the alternate segment theorem:** The angle between tangent $FDE$ and chord $AD$ at $D$ equals $y^\circ$. But this angle is also equal to the angle subtended by chord $BD$ at the circumference, which is $\frac{x}{2}^\circ$. 5. **Therefore:** $$ y = \frac{x}{2} $$ 6. **Find $y - x$:** $$ y - x = \frac{x}{2} - x = \frac{x}{2} - \frac{2x}{2} = -\frac{x}{2} $$ **Final answer:** $$ y - x = -\frac{x}{2}^\circ $$ This means $y$ is half of $x$, so $y - x$ is negative half of $x$.