The angle between the lines whose direction ratios satisfy the equations l+m+n=0 and $l^2=m^2+n^2$ 

$l+m+n=0\Rightarrow l=-(m+n)$

$l^2=m^2+n^2\Rightarrow {(m+n)}^2=m^2+n^2$

$\therefore m=0\left(or\right)n=0$

If m=0 then 

$l=-n\Rightarrow l:m:n=-n:0:n=(-1,0,1)$

If n=0 then 

$l=-m\Rightarrow l:m:n=-m:m:0=(-1,1,0)$

D.r’s of two lines are (-1,0,1) and (-1,1,0)

If ‘$\theta$’ is the angle between the lines then the 

$Cos\theta =\frac{(-1)(-1)+0\left(1\right)+1\left(0\right)}{\sqrt{1+0+1}\sqrt{1+1+0}}=\frac{1}{2}=\frac{\pi }{3}$