Angle $B$ is angle between $\overrightarrow {BA}$ and $\overrightarrow {BC}$
Given: $A(1,4,2),\:B(-2,1,2)\:C(2,-3,4)$
$\overrightarrow {BA}=(3,3,0)\:\:and\:\:\overrightarrow {BC}=(4,-4,2)$
Angle $B$ is given by $cos^{-1}\bigg(\large\frac{\overrightarrow {BA}.\overrightarrow {BC}}{|\overrightarrow {BA}||\overrightarrow {BC}|}\bigg)$
$=cos^{-1}\bigg(\large\frac{(3,3,0).(4,-4,2)}{\sqrt {18}.6}\bigg)$
$=cos^{-1}0=90^{\circ}$