1. The problem states that $\angle AOC$ is a central angle and intercepts arc $\widehat{AC}$. The measure of a central angle is equal to the measure of the intercepted arc.\n\n2. Given $m\angle AOC = 85^\circ$ and $m\widehat{AC} = 5x$. Because the central angle equals its intercepted arc, we have:\n$$ m\angle AOC = m\widehat{AC} $$\n$$ 85 = 5x $$\n\n3. Solve for $x$: \n$$ x = \frac{85}{5} = 17 $$\n\nThe value of $x$ is 17.