1. **Stating the problem:** We are given a geometric figure with angles $x$, $y$, and $z$ and the equation $x + y = 63^\circ$. We need to find the value of angle $z$. 2. **Understanding the problem:** The figure involves a circle passing through points $B$, $E$, and $C$. Angle $z$ is at point $B$ inside the circle, formed by segments $BE$ and $BC$. Angles $x$ and $y$ are related by $x + y = 63^\circ$. 3. **Key geometric rule:** In a circle, the measure of an angle formed by two chords intersecting inside the circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 4. **Applying the rule:** Angle $z$ at $B$ is an inscribed angle intercepting arc $EC$. Angles $x$ and $y$ are related to arcs intercepted by the circle. 5. **Using the given relation:** Since $x + y = 63^\circ$, and considering the properties of angles in the circle, angle $z$ equals half of $x + y$. 6. **Calculating $z$:** $$z = \frac{x + y}{2} = \frac{63^\circ}{2} = 31.5^\circ$$ 7. **Conclusion:** The value of angle $z$ is $31.5^\circ$.