Solution:

$For y=5 x+6 \quad \therefore 5 x-y+6=0$

$slope m_{1}=-\frac{a}{b}=-\frac{5}{-1}=5$

$For y=x \therefore x-y=0$

$slope m_{2}=-\frac{a}{b}=-\frac{1}{-1}=1$

$\begin{aligned} \therefore \tan \theta &=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right| \\ &=\left|\frac{5-1}{1+5 \times 1}\right| \\ \therefore \tan \theta &=\frac{2}{3} \\ \therefore \theta=\tan ^{-1}\left(\frac{2}{3}\right) \end{aligned}$