The pair of lines $x^2-xy-y^2$ and the pair of lines  $x^2+4xy-y^2=0$ are such that

The bisectors of the angles between the lines  $x^2-xy-y^2=0$ are given by

$\frac{x^2-y^2}{xy}=\frac{1-(-1)}{-\frac{1}{2}}=-4\left(\text{ or }\right)x^2+4xy-y^2=0$

Which is the same as the second pair, thus one of the
pair is bisecting the angles between the other pair.
Hence the four lines of the pairs are equally inclined
to one another at angles $\frac{\mathrm{\pi }}{4}$ .