Because $\displaystyle\angle{U}{V}{W}\usepackage{cmbright}\Huge\$ is a straight angle, it measures $\displaystyle{{180}}^{\circ}\usepackage{cmbright}\Huge\$ . The four angles along this straight angle have measures that total $\displaystyle{{180}}^{\circ}\usepackage{cmbright}\Huge\$ . So:

$\displaystyle{{43}}^{\circ}+{{52}}^{\circ}+{m}\angle{Y}{V}{Z}+{{38}}^{\circ}={{180}}^{\circ}\usepackage{cmbright}\Huge\$

$\displaystyle{m}\angle{Y}{V}{Z}={{180}}^{\circ}-{\left({{43}}^{\circ}+{{52}}^{\circ}+{{38}}^{\circ}\right)}={{47}}^{\circ}\usepackage{cmbright}\Huge\$