Let the direction ratios of the two lines be $a_1 = a, b_1 = b, c_1 = c$ and $a_2 = b-c, b_2 = c-a, c_2 = a-b$.

The angle $\theta$ between two lines with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ is given by the formula:

$$\cos \theta = \left| \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \right|$$

Step 1: Calculate the numerator. Substitute the direction ratios into the term $a_1 a_2 + b_1 b_2 + c_1 c_2$:

$$a(b-c) + b(c-a) + c(a-b)$$

Step 2: Simplify the expression.

$$= ab - ac + bc - ab + ac - bc$$ $$= (ab - ab) + (bc - bc) + (ac - ac)$$ $$= 0$$

Step 3: Determine the angle. Since the numerator is 0, we have:

$$\cos \theta = \left| \frac{0}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \right| = 0$$

As stated in the context, if $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$, the lines are perpendicular ($\theta = 90^\circ$).

Therefore, the angle between the lines is $90^\circ$.