\(SA\perp\left(ABCD\right)\Rightarrow SA\perp BD\)
Lại có \(BD\perp AC\) (hai đường chéo hv)
\(\Rightarrow BD\perp\left(SAC\right)\)
\(\left\{{}\begin{matrix}SA\perp\left(ABCD\right)\Rightarrow SA\perp BC\\BC\perp AB\left(gt\right)\end{matrix}\right.\) \(\Rightarrow BC\perp\left(SAB\right)\) (1)
\(\Rightarrow\widehat{CSB}\) là góc giữa SC và (SAB)
\(SB=\sqrt{SA^2+AB^2}=\frac{2a\sqrt{3}}{3}\)
\(\Rightarrow tan\widehat{CSB}=\frac{BC}{SB}=\frac{\sqrt{3}}{2}\Rightarrow\widehat{CSB}\approx41^0\)
\(\left\{{}\begin{matrix}SA\perp\left(ABCD\right)\Rightarrow SA\perp CD\\CD\perp AD\left(gt\right)\end{matrix}\right.\) \(\Rightarrow CD\perp\left(SAD\right)\) (2)
\(\Rightarrow\widehat{SDA}\) là góc giữa (SCD) và (ABCD)
\(tan\widehat{SDA}=\frac{SA}{AD}=\frac{\sqrt{3}}{3}\Rightarrow\widehat{SDA}=30^0\)
Từ (1) \(\Rightarrow BC\perp AH\), mà \(AH\perp SB\Rightarrow AH\perp\left(SBC\right)\Rightarrow AH\perp SC\) (3)
Từ (2) \(\Rightarrow CD\perp AK\), mà \(AK\perp SD\Rightarrow AK\perp\left(SCD\right)\Rightarrow AK\perp SC\) (4)
(3);(4) \(\Rightarrow SC\perp\left(AHK\right)\) \(\Rightarrow\left(SAC\right)\perp\left(AHK\right)\)
