\(AB=(5,-3,0)-(1,0,1)=(4,-3,1)\)

\(AC=(1,5,2)-(1,0,-1)=(0,5,3)\)

\(|AB|=\sqrt{4^2+3^2+1^2}=\sqrt{26}\)

\(|AC|=\sqrt{0+5^2+3^2}=\sqrt{34}\)

Use:

\(AB\cdot AC=|AB|\cdot|AC|\cos\angle CAB\)

\((4,-3,1)\cdot(0,5,3)=\sqrt{26}\cdot\sqrt{34}\cdot\cos\angle CAB\)

\(4\cdot0-3\cdot5+1\cdot3=\sqrt{26}\cdot\sqrt{34}\cdot\cos\angle CAB\)

\(-12=\sqrt{26}\cdot\sqrt{34}\cdot\cos\angle CAB\)

\(\cos\angle CAB=-0.404\)

\(\angle CAB=113.8\) degree

\(BA=(1,0,-1)-(5,-3,0)=(-4,3,-1)\)

\(BC=(1,5,2)-(5,-3,0)=(-4,8,2)\)

\(|BA|=\sqrt{26}\)

\(|BC|=\sqrt{4^2+8^2+2^2}=\sqrt{84}\)

Use:

\(BA\cdot BC=|BA|\cdot|BC|\cdot\cos\angle ABC\)

\((-4,3,-1)\cdot(-4,8,2)=\sqrt{26}\cdot\sqrt{84}\cdot\cos\angle ABC\)

\(16+24-2=\sqrt{26}\cdot\sqrt{84}\cdot\cos\angle ABC\)

\(\cos\angle ABC=0.813\)

\(\angle ABC=35.6\) degree

Sum of all \(\angle\) \(= 180\)

\(\angle BCA+35.6+113.8=180\)

\(\angle BCA=30.6\) degree