Trigonometric Functions · Superchallenge problem

Start with the point $(1,0)$. Reflect it across the line through the origin making an angle of $20^\circ$ with the positive $x$-axis, then reflect the result across the line through the origin at $50^\circ$, then across the $20^\circ$ line again, then the $50^\circ$ line, and so on, alternating. After $101$ reflections in all, the point is on the unit circle at angle $\alpha$ with $0\le\alpha<360$ (in degrees). Find $\alpha$.