Disclaimer: The solutions we've shared are just one exciting approach, and there are surely many other wonderful methods out there. We’d love to hear your alternative solutions in the community thread below, so let's keep the creativity flowing!

We can move the lines such that they all intersect in one point. Notice that, this does not change the angle between any pair of lines. After moving, there are $16$ small angles summing to $360^\circ$. So, by Pigeonhole Principle, one of these angles must be less than or equal to $\frac{360^\circ}{16}=22.5^\circ$, which is less than $23^\circ$ and we are done.