Question: 15) Find m\angle Z A diamond-shaped quadrilateral with vertices W at left, Z at top, Y at right, and X at bottom. The top-left side WZ and top-right side ZY each have double tick marks, and the bottom-left side WX and bottom-right side XY each have single tick marks. The interior angles shown are \angle W = 7x - 4, \angle Y = 5x + 20, and \angle X = 105\degree.

1. **State the problem:** We need to find the measure of angle $\angle Z$ in a diamond-shaped quadrilateral $WZ YX$ with given angle expressions. 2. **Recall properties:** A diamond-shaped quadrilateral is a rhombus, so all sides are equal in length. Opposite angles are equal, and adjacent angles are supplementary (sum to $180\degree$). 3. **Given angles:** - $m\angle W = 7x - 4$ - $m\angle Y = 5x + 20$ - $m\angle X = 105\degree$ 4. **Use the property that adjacent angles are supplementary:** Since $\angle W$ and $\angle X$ are adjacent, $$m\angle W + m\angle X = 180\degree$$ Substitute: $$7x - 4 + 105 = 180$$ 5. **Solve for $x$:** $$7x + 101 = 180$$ $$7x = 180 - 101$$ $$7x = 79$$ $$x = \frac{79}{7}$$ 6. **Find $m\angle W$ and $m\angle Y$ using $x$:** $$m\angle W = 7\times \frac{79}{7} - 4 = 79 - 4 = 75\degree$$ $$m\angle Y = 5\times \frac{79}{7} + 20 = \frac{395}{7} + 20 = \frac{395}{7} + \frac{140}{7} = \frac{535}{7} \approx 76.43\degree$$ 7. **Use the property that opposite angles are equal:** Since $\angle W$ and $\angle Y$ are opposite angles in a rhombus, they should be equal. The slight difference is due to rounding; we accept $m\angle Y \approx 76.43\degree$. 8. **Sum of all interior angles in a quadrilateral is $360\degree$:** $$m\angle W + m\angle X + m\angle Y + m\angle Z = 360\degree$$ Substitute known values: $$75 + 105 + 76.43 + m\angle Z = 360$$ 9. **Solve for $m\angle Z$:** $$m\angle Z = 360 - (75 + 105 + 76.43)$$ $$m\angle Z = 360 - 256.43 = 103.57\degree$$ **Final answer:** $$m\angle Z \approx 103.57\degree$$