Answer
$45^{\circ}$ or $\pi/4$
Work Step by Step
The result of exercise 31 tells us that
${\bf v}=\langle a,\ b \rangle$ is perpendicular to lines $ax+by=c$
${\bf n_{1}}=\langle 1,\sqrt{3}\rangle$ is perpendicular to $x+\sqrt{3}y=1$
${\bf n_{2}}=\langle 1-\sqrt{3},1+\sqrt{3}\rangle$ is perpendicular to $(1-\sqrt{3})x+(1+\sqrt{3})y=8.$
$\displaystyle \theta=\cos^{-1}(\frac{{\bf n_{1}}\cdot{\bf n_{2}}}{|{\bf n_{1}}||{\bf n_{2}}|})$
$=\displaystyle \cos^{-1}(\frac{(1)(1-\sqrt{3})+\sqrt{3}(1+\sqrt{3})}{\sqrt{1+3}\cdot\sqrt{(1-\sqrt{3})^{2}+(1+\sqrt{3})}})$
$=\displaystyle \cos^{-1}(\frac{1-\sqrt{3}+\sqrt{3}+3}{2\cdot\sqrt{1-2\sqrt{3}+3+1+2\sqrt{3}+3}})$
$=\displaystyle \cos^{-1}(\frac{4}{2\cdot\sqrt{8}})$
$=\displaystyle \cos^{-1}(\frac{1}{\sqrt{2}})=45^{\circ}$ or $\pi/4$
