1. **Stating the problem:** Given that $\angle AOB = 125^\circ$, find the measure of $\angle COD$ in the figure where points A, B, C, D, and O form a pentagon with a circle centered at O. 2. **Understanding the geometry:** Since O is the center of the circle and points A, B, C, D lie on the circle, $\angle AOB$ and $\angle COD$ are central angles. 3. **Key property:** Central angles that subtend arcs on a circle add up to 360° because the full circle is 360°. 4. **Using the property:** The angles around point O must sum to 360°, so $$\angle AOB + \angle BOC + \angle COD + \angle DOA = 360^\circ$$ 5. **Assuming $\angle BOC$ and $\angle DOA$ are equal or not given, but since the problem only asks for $\angle COD$ and the figure is symmetric or the angles are opposite, $\angle COD$ is the vertical angle to $\angle AOB$. 6. **Vertical angles are equal:** Therefore, $$\angle COD = \angle AOB = 125^\circ$$ **Final answer:** $\boxed{125^\circ}$