C

To find the angle \( \theta \) between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \), we can use the property that the resultant of the two vectors is perpendicular to another vector. Let the magnitudes of \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) be \( A \) and \( B \), respectively. When the resultant \( \overrightarrow{R} = \overrightarrow{OA} + \overrightarrow{OB} \) is perpendicular to the vector \( \overrightarrow{OC} \), the following condition holds: \( A^2 + B^2 + 2AB \cos(\theta) = 0 \).
Rearranging gives: \( \cos(\theta) = -\frac{A^2 + B^2}{2AB} \).
Therefore, angle \( \theta = \cos^{-1}\left( -\frac{A^2 + B^2}{2AB} \right) \). Given this relation results in an angle associated with a specific condition of perpendicularity, the solution will provide a specific angle value based upon the magnitudes of the vectors. Thus, in this context, the angle \( \theta \) that also satisfies the given conditions leads to the conclusion that this angle is indeed sufficiently explained by option C.