1. **Problem Statement:** We are given a circle with center $O$ and a tangent line $AC$ touching the circle at point $C$. We need to find the measure of angle $\angle OAC$, which is given as 63°. 2. **Key Concept:** The radius drawn to the point of tangency is perpendicular to the tangent line. This means $\angle OCA = 90^\circ$ because $OC$ is a radius and $AC$ is tangent at $C$. 3. **Given:** $\angle OAC = 63^\circ$ (the angle we want to confirm or understand in context). 4. **Triangle $OAC$:** Since $OC$ is perpendicular to $AC$, triangle $OAC$ is a right triangle with right angle at $C$. 5. **Sum of angles in triangle:** The sum of angles in triangle $OAC$ is $180^\circ$. So, $$\angle OAC + \angle ACO + \angle COA = 180^\circ$$ 6. **Substitute known values:** $\angle ACO = 90^\circ$, $\angle OAC = 63^\circ$, so $$63^\circ + 90^\circ + \angle COA = 180^\circ$$ 7. **Calculate $\angle COA$:** $$\angle COA = 180^\circ - 63^\circ - 90^\circ = 27^\circ$$ 8. **Conclusion:** The measure of $\angle OAC$ is indeed $63^\circ$ as given, consistent with the properties of the tangent and radius. Thus, $\boxed{63^\circ}$ is the measure of $\angle OAC$.