$M$ is the midpoint of $AC$ and $BD$

$M\left(\frac{3+7}{2},\frac{2+4}{2},\frac{3+5}{2}\right)=M\left(5,3,4\right)$

Now using diagonal $BD$

$\frac{x+1}{2}=5 \Rightarrow x=9$

$\frac{y+4}{2}=3\Rightarrow y=2$

$\frac{z+6}{2}=4\Rightarrow z=2$

Direction ratios of $DC, a_1=2, b_1=-2, c_1=-3$

Direction ratios of $DB, a_2=8, b_2=-2, c_2=-4$

Angle $\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2}+c_2^2}$

$\Rightarrow \cos \theta =\frac{32}{\sqrt{84}\sqrt{17}}=\frac{16}{\sqrt{21}\sqrt{17}}=\frac{16}{\sqrt{357}}$

$\Rightarrow \theta ={\cos }^{-1}\left(\frac{16}{\sqrt{357}}\right)$