1. **State the problem:** We have a triangle with sides 5 km, 12 km, and 13 km. We need to find the greatest angle between the roads. 2. **Identify the greatest angle:** The greatest angle is opposite the longest side. Here, the longest side is 13 km. 3. **Use the Law of Cosines:** For a triangle with sides $a$, $b$, and $c$, and angle $C$ opposite side $c$, the law states: $$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$ 4. **Substitute values:** Let $a=5$, $b=12$, and $c=13$. $$\cos C = \frac{5^2 + 12^2 - 13^2}{2 \times 5 \times 12} = \frac{25 + 144 - 169}{120} = \frac{0}{120} = 0$$ 5. **Calculate angle $C$:** Since $\cos C = 0$, then $$C = \cos^{-1}(0) = 90^\circ$$ 6. **Conclusion:** The greatest angle between the roads is $90^\circ$, meaning the roads form a right angle.