= Solution
Choose the interface as $x=0$ and write the incident, reflected, and transmitted fields as plane waves with wavevectors $k_I,k_R,k_T$. The boundary conditions must hold for every tangential coordinate and every time. The <phase matching at a planar wave interface> therefore requires
$$
\omega_I=\omega_R=\omega_T
$$
and equality of tangential wavevectors:
$$
(k_I)_\parallel=(k_R)_\parallel=(k_T)_\parallel.
$$
The incident and reflected waves are in the same medium, so
$|k_I|=|k_R|$. Equality of their tangential components gives
$$
|k_I|\sin\theta_I=|k_R|\sin\theta_R,
$$
hence
$$
\boxed{\theta_R=\theta_I}.
$$
At fixed frequency, $|k_\pm|=\omega/v_\pm$ is proportional to the refractive index $n_\pm$. Tangential phase matching between incident and transmitted waves gives the <Snell law for electromagnetic waves>:
$$
|k_-|\sin\theta_I=|k_+|\sin\theta_T,
$$
or
$$
\boxed{n_-\sin\theta_I=n_+\sin\theta_T}.
$$
Here $n_\pm=c/v_\pm=c\sqrt{\epsilon_\pm\mu}$ if the vacuum speed $c$ is used to normalize the index.
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