Question:

To form a parallelogram, we need 2 lines from one set and 2 lines from another set. With two sets of $m$ parallel lines each, the number of ways to choose 2 lines from the first set is $\binom{m}{2}$ and from the second set is also $\binom{m}{2}$. The total number of parallelograms formed is $\binom{m}{2} \times \binom{m}{2} = 225$. This gives us $\left(\frac{m(m-1)}{2}\right)^2 = 225$, so $\frac{m(m-1)}{2} = 15$. Solving: $m(m-1) = 30$, which yields $m^2 - m - 30 = 0$, factoring as $(m-6)(m+5) = 0$. Since $m$ must be positive, $m = 6$.