1. **State the problem:** We have parallelogram VWXY with diagonals WY and VX intersecting at Z. Given segment WZ = 9, angle X = 64°, angle Y = 39°, and segment ZY = 4x + 1, find $x$, $m\angle ZVW$, and $m\angle ZWV$. 2. **Recall properties of parallelograms:** Opposite sides are equal and parallel. Diagonals bisect each other, so $WZ = ZY$. 3. **Use diagonal bisection:** Since diagonals bisect each other, $WZ = ZY$. Given $WZ = 9$ and $ZY = 4x + 1$, set equal: $$9 = 4x + 1$$ 4. **Solve for $x$:** $$9 - 1 = 4x$$ $$8 = 4x$$ $$x = \frac{8}{4} = 2$$ 5. **Find $m\angle ZVW$ and $m\angle ZWV$:** - Angles at vertices $V$ and $W$ are adjacent to angles $X$ and $Y$. - In parallelograms, consecutive angles are supplementary: $$m\angle V + m\angle W = 180^\circ$$ 6. **Calculate $m\angle V$ and $m\angle W$:** Given $m\angle X = 64^\circ$ and $m\angle Y = 39^\circ$, opposite angles are equal: $$m\angle V = m\angle X = 64^\circ$$ $$m\angle W = m\angle Y = 39^\circ$$ 7. **Find $m\angle ZVW$ and $m\angle ZWV$ inside triangles:** - Triangle $VZW$ is formed by points $V$, $Z$, and $W$. - Since $Z$ is midpoint of diagonal, $VZ = ZX$ and $WZ = ZY$. - Use triangle angle sum: $$m\angle ZVW + m\angle ZWV + m\angle VZW = 180^\circ$$ 8. **Calculate $m\angle VZW$:** - $m\angle VZW$ is angle at $Z$ between $V$ and $W$. - Since $Z$ is intersection of diagonals, $m\angle VZW = 180^\circ - m\angle X = 180^\circ - 64^\circ = 116^\circ$ 9. **Calculate $m\angle ZVW$ and $m\angle ZWV$:** $$m\angle ZVW + m\angle ZWV = 180^\circ - 116^\circ = 64^\circ$$ 10. **Assuming triangle $VZW$ is isosceles (since $VZ = WZ$), angles opposite equal sides are equal:** $$m\angle ZVW = m\angle ZWV = \frac{64^\circ}{2} = 32^\circ$$ **Final answers:** $$x = 2$$ $$m\angle ZVW = 32^\circ$$ $$m\angle ZWV = 32^\circ$$