1. **State the problem:** We have a rectangle ABCD with diagonals AC and BD intersecting at point E. Given that $BD = 4x - 2$, $AC = 5x - 10$, and $m\angle EDA = 39^\circ$, find the length $AE$ and the measure $m\angle EAB$. 2. **Recall properties of rectangles and diagonals:** In a rectangle, the diagonals are equal in length and bisect each other. This means: $$BD = AC$$ and $$AE = EC = BE = ED = \frac{AC}{2} = \frac{BD}{2}$$ 3. **Set up the equation for the diagonals:** $$4x - 2 = 5x - 10$$ 4. **Solve for $x$:** $$4x - 2 = 5x - 10$$ $$-2 + 10 = 5x - 4x$$ $$8 = x$$ 5. **Find the length of diagonal $AC$ (or $BD$):** $$AC = 5x - 10 = 5(8) - 10 = 40 - 10 = 30$$ 6. **Find $AE$ (half of diagonal $AC$):** $$AE = \frac{AC}{2} = \frac{30}{2} = 15$$ 7. **Find $m\angle EAB$:** Since $E$ is the midpoint of the diagonals, and $m\angle EDA = 39^\circ$, note that $\angle EDA$ and $\angle EAB$ are congruent because opposite angles formed by intersecting diagonals in a rectangle are equal. Therefore: $$m\angle EAB = 39^\circ$$ **Final answers:** $$AE = 15$$ $$m\angle EAB = 39^\circ$$