Recall that the opposite angles of an inscribed quadrilateral are supplementary. Therefore, an external angle of an inscribed quadrilateral has the same measure as the opposite angle of the inscribed quadrilateral.
Extend CF and DE to intersect at G. Let the angle bisector of CGD intersect FE at H and CD at K.
Consider the angles EHK and DKH in terms of x and z.
The bisector of the vertex angle of an isosceles triangle is perpendicular to the base.