Elementary Geometry for College Students (5th Edition)

m$\angle$2=m$\angle$3=48$^{\circ}$ and m$\angle$5=m$\angle$6=42$^{\circ}$
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}$