Explanation

(a)
Angle between the faces $OAB$ and $ABC$ is same as angle between normals of faces $OAB$ and $ABC$.
Vector along the normals of $OAB$
$= \left| \begin{matrix}
\widehat{\mathbf{i}} & \widehat{\mathbf{j}} & \widehat{\mathbf{k}} \\
1 & 2 & 1 \\
2 & 1 & 3
\end{matrix} \right| = 5\widehat{\mathbf{i}} - \widehat{\mathbf{j}} - 3\widehat{\mathbf{k}} = \overrightarrow{\mathbf{a}}$ (let)
Vector along normals of $ABC$
$= \left| \begin{matrix}
\widehat{\mathbf{i}} & \widehat{\mathbf{j}} & \widehat{\mathbf{k}} \\
1 & - 1 & 2 \\
- 2 & - 1 & 1
\end{matrix} \right| = \widehat{\mathbf{i}} - 5\widehat{\mathbf{j}} - 3\widehat{\mathbf{k}} = \overrightarrow{\mathbf{b}}$ (let)
$\therefore\cos{\theta = \frac{\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}}{\left| \overrightarrow{\mathbf{a}} \right|\left| \overrightarrow{\mathbf{b}} \right|} = \frac{5 + 5 + 9}{\sqrt{35}\sqrt{35}}}$
$\Rightarrow \ \theta = \cos^{- 1}\left( \frac{19}{35} \right)$