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.