 Solution

Take the positive horizontal direction along B. The initial direction is $(\alpha -90^{\circ })-\theta$. Reflection at the horizontal face B reverses the vertical direction. At C, symmetry of the equilateral geometry gives internal incidence $r$, so Snell's law makes the emerging angle to C's normal equal to $\theta$. Its direction is therefore $(90^{\circ }-\alpha )+\theta$. Subtracting the incoming direction givesThis is the smaller angle between the incident and emergent propagation directions in the requested reflection interval. The cancellation of $n$ occurs because the entry and exit refraction angles have equal magnitudes; refractive index still determines whether the reflected route exists and where it exits.