Consider the picture where a particle moves inside a closed disk and reflects specularly from the boundary as per the angle of incidence equals the angle of reflection rule. This will probably be a trivial question, but by what geometric tools can you argue that $\alpha_0 = \alpha_1$? The particle starts at the point $\varphi_0$ with an initial angle of $\alpha_0$ to the circle's tangent line at $\varphi_0$. The point $\varphi_1$ is given by $\varphi_1 = \varphi_0 + \vartheta \mod 2\pi$.
