1. **Stating the problem:** We have a diamond-shaped quadrilateral ABCD with sides labeled as follows: - AD = 3y - DC = 2x + 3 - BC = 2y - 2 - AB = \sqrt{5} at 45° Given AC and DB diagonals for two cases: Case 1: AC = 10, DB = 17, find A. Case 2: AC = 16, DB = 30, find A. 2. **Important properties of a diamond (rhombus):** - All sides are equal in length. - Diagonals bisect each other at right angles. - Opposite angles are equal. 3. **Using side equality:** Since ABCD is a diamond, all sides are equal: $$3y = 2x + 3 = 2y - 2 = AB$$ 4. **Equate sides to find relations:** From $3y = 2y - 2$: $$3y = 2y - 2$$ $$3y - 2y = -2$$ $$y = -2$$ From $3y = 2x + 3$: $$3(-2) = 2x + 3$$ $$-6 = 2x + 3$$ $$2x = -9$$ $$x = \frac{\cancel{-9}}{\cancel{2}}$$ 5. **Check side length AB:** $$AB = 3y = 3(-2) = -6$$ Side length cannot be negative, so $y = -2$ is invalid. 6. **Re-examine side equality:** Try $2x + 3 = 2y - 2$: $$2x + 3 = 2y - 2$$ $$2x - 2y = -5$$ Try $3y = 2x + 3$: $$3y = 2x + 3$$ 7. **Solve system:** From $3y = 2x + 3$: $$2x = 3y - 3$$ $$x = \frac{3y - 3}{2}$$ Substitute into $2x - 2y = -5$: $$2\left(\frac{3y - 3}{2}\right) - 2y = -5$$ $$3y - 3 - 2y = -5$$ $$y - 3 = -5$$ $$y = -2$$ Again $y = -2$ invalid. 8. **Since side lengths must be positive, check if AB = \sqrt{5} is the side length:** Given AB = \sqrt{5}, set all sides equal to \sqrt{5}: $$3y = \sqrt{5}$$ $$y = \frac{\sqrt{5}}{3}$$ $$2x + 3 = \sqrt{5}$$ $$2x = \sqrt{5} - 3$$ $$x = \frac{\sqrt{5} - 3}{2}$$ $$2y - 2 = \sqrt{5}$$ $$2y = \sqrt{5} + 2$$ $$y = \frac{\sqrt{5} + 2}{2}$$ 9. **Check consistency of y values:** From 8, two different y values: $$y = \frac{\sqrt{5}}{3}$$ and $$y = \frac{\sqrt{5} + 2}{2}$$ Not equal, so sides are not equal unless variables satisfy these. 10. **Use diagonals to find angle A:** In a rhombus, diagonals intersect at right angles and bisect each other. Using diagonal lengths and side length $s = AB = \sqrt{5}$, angle $A$ can be found by: $$\cos A = \frac{d_1^2 + d_2^2}{2 \times s^2} - 1$$ where $d_1 = AC$, $d_2 = DB$. 11. **Case 1: AC = 10, DB = 17, $s = \sqrt{5}$** $$\cos A = \frac{10^2 + 17^2}{2 \times (\sqrt{5})^2} - 1 = \frac{100 + 289}{2 \times 5} - 1 = \frac{389}{10} - 1 = 38.9 - 1 = 37.9$$ Since $\cos A$ cannot be greater than 1, this is invalid. 12. **Case 2: AC = 16, DB = 30, $s = \sqrt{5}$** $$\cos A = \frac{16^2 + 30^2}{2 \times 5} - 1 = \frac{256 + 900}{10} - 1 = \frac{1156}{10} - 1 = 115.6 - 1 = 114.6$$ Again invalid. 13. **Conclusion:** Given the side lengths and diagonals, the problem data is inconsistent for a diamond with side length $\sqrt{5}$. More information or correction is needed. **Final answer:** Cannot determine angle A with given inconsistent data.