1. **Problem statement:** Given that line $\overleftrightarrow{AC}$ is tangent to circle $O$ at point $C$, and $\angle ACB = 28^\circ$, find the measure of $\angle ACO$. 2. **Key fact:** The radius drawn to a tangent point is perpendicular to the tangent line. This means $\angle ACO = 90^\circ$ because $OC$ is a radius and $AC$ is tangent at $C$. 3. **Explanation:** Since $AC$ is tangent at $C$, $OC$ is perpendicular to $AC$. Therefore, $\angle ACO = 90^\circ$. 4. The given $28^\circ$ angle is not directly related to $\angle ACO$ in this context. The tangent-radius perpendicularity rule gives the answer immediately. **Final answer:** $$\angle ACO = 90^\circ$$