Properties of Isosceles Trapezoids

Suppose $ABCD$ is an isosceles trapezoid where $\overline{AB} \parallel \overline{CD}$ and the angle measures in degrees are represented in terms of variables $x, y, z$ as follows:

$$\begin{align} m\angle CAB &= x + y - z \\ m\angle ABD &= 2x + z + 20 \\ m\angle BDC &= \frac{x + y}{2} \\ m\angle DCA &= y - z - 10. \end{align}$$

Find the value of $m\angle BDC.$