jmc

Suppose that $\angle ADC=x^{\circ }$. The area of the unshaded portion of the inner circle is thus $\frac{x}{360}$ of the total area of the inner circle, or $\frac{x}{360}(\pi (1^2))=\frac{x}{360}\pi$ (since $\angle ADC$ is $\frac{x}{360}$ of the largest possible central angle ($360^{\circ }$)).

The area of shaded portion of the inner circle is thus $$\pi -\frac{x}{360}\pi =\frac{360-x}{360}\pi .$$ The total area of the outer ring is the difference of the areas of the outer and inner circles, or $\pi (2^2)-\pi (1^2)=3\pi$. The shaded region of the outer ring will be $\frac{x}{360}$ of this total area. So the shaded region of the outer ring is $\frac{x}{360}(3\pi )=\frac{3x}{360}\pi$.

So the total shaded area (which must equal $\frac{5}{3}\pi$) is, in terms of $x$, $$\frac{3x}{360}\pi +\frac{360-x}{360}\pi =\frac{360+2x}{360}\pi .$$ Therefore, $$\frac{360+2x}{360}=\frac{5}{3}=\frac{600}{360},$$ so $360+2x=600$, or $x=\boxed{120}$.