1. **Problem statement:** We have kite $WXYZ$ with angles $m\angle W = 74^\circ$ and $m\angle Y = 56^\circ$. We need to find $m\angle Z$. 2. **Recall kite properties:** In a kite, two pairs of adjacent sides are equal. Also, the sum of interior angles in any quadrilateral is $360^\circ$. 3. **Sum of angles formula:** For quadrilateral $WXYZ$, $$m\angle W + m\angle X + m\angle Y + m\angle Z = 360^\circ$$ 4. **Substitute known values:** $$74^\circ + m\angle X + 56^\circ + m\angle Z = 360^\circ$$ 5. **Simplify known angles:** $$130^\circ + m\angle X + m\angle Z = 360^\circ$$ 6. **Use kite property about angles between unequal sides:** Angles $W$ and $Y$ are between unequal sides, so angles $X$ and $Z$ are equal. Thus, $$m\angle X = m\angle Z$$ 7. **Replace $m\angle X$ with $m\angle Z$ in equation:** $$130^\circ + m\angle Z + m\angle Z = 360^\circ$$ 8. **Combine like terms:** $$130^\circ + 2m\angle Z = 360^\circ$$ 9. **Isolate $m\angle Z$:** $$2m\angle Z = 360^\circ - 130^\circ$$ $$2m\angle Z = 230^\circ$$ 10. **Divide both sides by 2:** $$m\angle Z = \frac{230^\circ}{2}$$ $$m\angle Z = 115^\circ$$ **Final answer:** $$m\angle Z = 115^\circ$$