In a figure, three lines intersect at point P forming adjacent angles $x^{\circ }$, $y^{\circ }$, and $z^{\circ }$ on a straight line such that the angle between the $y^{\circ }$ and $z^{\circ }$ angles is vertical to the $x^{\circ }$ angle. If $x=50$ and $y=60$, what is the value of $z$?