The cute  angle between the lines whose direction ratios are given by $l + m - n = 0 $ and $ l^2 + m^2 - n^2 = 0 , $ is

$\frac{\pi}{3}$

We have,

$l + m - n = 0 $ and $ l^2 + m^2 - n^2 = 0 $

$⇒ l + m = n $ and $ l^2 + m^2 = n^2 $

$⇒ l^2 + m^2 = (l + m)^2 $        {On eliminating n]

$⇒ 2lm = 0 ⇒ l = 0 $ or , $ m = 0 $.

IF $ l = 0, $ then

$l + m - n = 0 ⇒ m = n $ $∴ \frac{l}{0}=\frac{m}{1}=\frac{n}{1}$

If $ m= 0 $, then 

$ l + m - n = 0 ⇒ l = n $∴ $\frac{l}{1}=\frac{m}{0}=\frac{n}{1}$

Thus, the direction ratios of the lines are proportional to 0, 1, 1 and 1, 0, 1. Therefore, angle $\theta $ between them is given by 

$ cos \theta = \frac{0×1+1×0+1×1}{\sqrt{0+1+1}\sqrt{1+0+1}}=\frac{1}{2}⇒\theta = \frac{\pi}{3}$