\[\begin{array}{l} {\text{а)}} \hfill \\ A{D^2} + S{A^2} = {6^2} + {\sqrt {21} ^2} = 57 = S{D^2} \Rightarrow \angle SAD = \frac{\pi }{2}. \hfill \\ A{B^2} + S{A^2} = {8^2} + {\sqrt {21} ^2} = 85 = S{B^2} \Rightarrow \angle SAB = \frac{\pi }{2} \hfill \\ \left. \begin{array}{l} SA \bot AD \hfill \\ SA \bot AB \hfill \\ \end{array} \right\} \Rightarrow SA \bot \left( {ABC} \right) \Rightarrow SA{\text{ - высота пирамиды }}SABCD. \hfill \\ {\text{б)}} \hfill \\ FE{\text{ - средняя линия }}\vartriangle ASC \Rightarrow FE\parallel SC \Rightarrow \alpha = \angle FED. \hfill \\ AC = \sqrt {A{D^2} + C{D^2}} = \sqrt {{6^2} + {8^2}} = 10. \hfill \\ SC = \sqrt {A{C^2} + S{A^2}} = \sqrt {100 + 21} = 11. \hfill \\ FE = \frac{1}{2}SC = \frac{{11}}{2}. \hfill \\ DE = \frac{1}{2}AC = 5. \hfill \\ DF = \sqrt {A{D^2} + A{F^2}} = \sqrt {{6^2} + {{\left( {\frac{{\sqrt {21} }}{2}} \right)}^2}} = \frac{{\sqrt {165} }}{2}. \hfill \\ {\text{Теорема косинусов для }}\vartriangle DFE: \hfill \\ D{F^2} = D{E^2} + F{E^2} - 2 \cdot DE \cdot FE \cdot \cos \alpha \Leftrightarrow \hfill \\ \cos \alpha = \frac{{{5^2} + {{\left( {\frac{{11}}{2}} \right)}^2} - \frac{{165}}{4}}}{{2 \cdot 5 \cdot \frac{{11}}{2}}} \Leftrightarrow \cos \alpha = \frac{{14}}{{55}} \Leftrightarrow \alpha = \arccos \frac{{14}}{{55}}. \hfill \\ \end{array}\]

б) \[\arccos \frac{{14}}{{55}}\]