1. The problem asks to estimate the angle of a green ray on the unit circle that starts at the origin and points to a position in Quadrant III, slightly below the negative x-axis, within 10 degrees. 2. Recall that the unit circle is centered at the origin $(0,0)$, and angles are measured from the positive x-axis counterclockwise. 3. Quadrant III corresponds to angles between $180^\circ$ and $270^\circ$. 4. Since the ray is slightly below the negative x-axis, the angle is just a bit more than $180^\circ$. 5. Estimating within 10 degrees, the angle is approximately $190^\circ$. 6. Therefore, the green ray's angle is about $190^\circ$ from the positive x-axis. Final answer: $$\boxed{190^\circ}$$