Tiling of regular polygon with similar right triangles

A tiling is a decomposition of a polygon into finitely many non-overlapping triangles. We prove that if a regular n-gon, $n \geq 5$, $n \neq 28$, can be tiled with similar right triangles, then one of the angles of these triangles is in $\left\{\frac{\pi}{n},\frac{2\pi}{n}, \frac{\pi} {6}+\frac{2\pi}{3n}\right\}$. Some related results were previously obtained by M.Laczkovich and B. Szegedy.