Answer
m$\angle$2=m$\angle$3=48$^{\circ}$
and
m$\angle$5=m$\angle$6=42$^{\circ}$
Work Step by Step
Given $\overline{UW}$$\parallel$$\overline{XZ}$
$\overline{VY}$$\bot$$\overline{UW}$
and $\overline{VY}$$\bot$$\overline{XZ}$
Also it is given that m$\angle$1=m$\angle$4=42$^{\circ}$
Therefore, m$\angle$2=m$\angle$3 because they are bisected by perpendicular line.
Therefore m$\angle$1+m$\angle$2=90$^{\circ}$
m$\angle$2=90$^{\circ}$-42$^{\circ}$
m$\angle$2=m$\angle$3=48$^{\circ}$
Now according to given properties of figure it is clear that
m$\angle$1=m$\angle$5 because both are alternate angles.
That means m$\angle$4=m$\angle$6
and we are given m$\angle$1=m$\angle$4=42$^{\circ}$
Therefore,
m$\angle$1=m$\angle$4=m$\angle$5=m$\angle$6=42$^{\circ}$
