TSTST 2020 Problem 3

We say a nondegenerate triangle whose angles have measures $\theta_{1}, \theta_{2}, \theta_{3}$ is quirky if there exists integers $r_{1}, r_{2}, r_{3}$, not all zero, such that $$r_{1} \theta_{1}+r_{2} \theta_{2}+r_{3} \theta_{3}=0$$ Find all integers $n \geq 3$ for which a triangle with side lengths $n-1, n, n+1$ is quirky.