Answer
a) $\theta$ will increase
b) $\theta=23.74^{\circ}$
Work Step by Step
(a) We know that
$sin\theta=\frac{n_{glass}}{n_{air}}sin\theta_r$. This equation shows that if the angle of refraction $\theta _r$ is constant then the angle of incidence $\theta$ is directly proportional to the refractive index of glass $n_{glass}$. Thus if $n_{glass}$ increases, so will $\theta$.
(b) We can find the required angle as follows:
$\theta =\frac{n_{glass}}{n_{air}}sin\theta_r$
but $\theta _r=tan^{-1}(\frac{5.00cm}{20.0cm})$
$\implies \theta =\frac{n_{glass}}{n_{air}}sin\space (tan^{-1}(\frac{5.00cm}{20.0cm}))$
We plug in the known values to obtain:
$\theta=sin^{-1}[\frac{1.66}{1.00}sin(tan^{-1}(\frac{5.00}{20.00}))]$
$\theta=23.74^{\circ}$
