1. We are given several geometric diagrams involving circles and angles, and we need to find the variable $z$ in each. 2. For the first diagram, angles of 48° are given at points on the circle involving vertices with variables $x$, $z$, and $y$. 3. Since the angles at points on a circle inscribed in the same arc are equal, applying the properties of cyclic quadrilaterals and inscribed angles allows us to find relationships between $x$, $y$, and $z$. 4. Specifically, in the first diagram, if two inscribed angles intercept the same arc, then they are equal, so $z=48°$. 5. In the second diagram, the angle 24° at point $A$ and variables $x$, $y$, $z$ on segments can be used to set up equations using the inscribed angle theorem and the sum of angles in cyclic quadrilaterals. 6. Using the property that opposite angles of a cyclic quadrilateral sum to 180°, and the fact that the orange shaded triangle and yellow shaded quadrilateral are inscribed in the circle, we find equations involving $x$, $y$, $z$. 7. Solving for $z$ in this setup yields $z = 24°$. 8. For the third diagram, angles of 42°, 30°, and 300° are given with variables $x$, $y$, $z$. 9. Note that 300° corresponds to reflex angles inside the circle. 10. Using the fact that the central angle equals the measure of the arc it intercepts, and the inscribed angles relate to arcs, we get equations relating $x$, $y$, $z$. 11. Applying these along with angle sum properties in triangles and the circle, we find $z = 48°$ in this diagram. **Final answers:** - In the first diagram, $z = 48°$ - In the second diagram, $z = 24°$ - In the third diagram, $z = 48°$