Show that the angles satisfy $x+y=z$
Solution 1:
(This space intentionally left blank.)
Solution 2:
Hint: denoting the points like on the picture below, triangles $EGD$ and $DGF$ are similar (why?).
Solution 3:
Let $X$, $Y$, and $Z$ be the apexes of the angles $x$, $y$, and $z$, respectively. Also, let $P$ be the common intersection of the red lines. Show that $ZP^2=ZX\cdot ZY$. Thus, the circumcircle of the triangle $PXY$ is tangent to $PZ$ at $P$. This will prove that $\angle YPZ=\angle YXP=x$.