imc

Let the area of the shaded region be $S$, the area of the unshaded region be $U$, and the acute angle that is formed by the two lines be $\theta$. We can set up two equations between $S$ and $U$: $S+U=9\pi$ $S=\frac{8}{13}U$ Thus $\frac{21}{13}U=9\pi$, and $U=\frac{39\pi }{7}$, and thus $S=\frac{8}{13}\cdot \frac{39\pi }{7}=\frac{24\pi }{7}$. Now we can make a formula for the area of the shaded region in terms of $\theta$: $\frac{2\theta }{2\pi }\cdot \pi +\frac{2(\pi -\theta )}{2\pi }\cdot (4\pi -\pi )+\frac{2\theta }{2\pi }(9\pi -4\pi )=\theta +3\pi -3\theta +5\theta =3\theta +3\pi =\frac{24\pi }{7}$ Thus $3\theta =\frac{3\pi }{7}\Rightarrow \theta =\frac{\pi }{7}\Rightarrow \boxed{\frac{\pi }{7}}$