A light ray is incident perpendicular to one face of a $90^{\circ }$ prism and is totally internally reflected at the glass-air interface. If the angle of reflection is $45^{\circ }$, we conclude that the refractive index $n$ is:

The light enters perpendicular, so it goes straight to the hypotenuse. The angle of incidence $i$ on the glass-air interface is $45^{\circ }$.For Total Internal Reflection (TIR) to occur, the angle of incidence must be greater than the critical angle $C$.
$$i>C⟹\sin i>\sin C$$
Given $i=45^{\circ }$ and $\sin C=1/n$:
$$\sin 45^{\circ }>\frac{1}{n}⟹\frac{1}{\sqrt{2}}>\frac{1}{n}$$
$$n>\sqrt{2}$$