Given the geometry of the isosceles right triangle $\triangle ABC$ with the conditions provided, we can use several geometric principles to solve for the length of $AD$.

Step-by-Step Solution:

1. Understand the Geometry:

   • Since $\triangle ABC$ is an isosceles right triangle, we know: $AC = BC$ and $\angle C = 90^\circ$.

   • Points $D$ and $E$ lie on sides $AC$ and $BD$ respectively, and the angle $\angle AED = 45^\circ$.

2. Label the Sides:

   • Let $AC = BC = x$, as $\triangle ABC$ is isosceles.
   • From the Pythagorean theorem in $\triangle ABC$, we know the length of the hypotenuse $AB$ is: $AB = \sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2}.$

3. Analyze the Given Lengths:

   • $AE = 4$, and $\angle AED = 45^\circ$, which means $\triangle AED$ forms an isosceles right triangle. Hence, $AD = AE = 4$, because in an isosceles right triangle, the two legs are congruent.

4. Use the Geometry of Triangle $\triangle AED$:

   • From the information that $\angle AED = 45^\circ$ and $AE = 4$, we conclude that $AD = 4$, as explained earlier.

Thus, the length of $AD$ is 4 units.

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Questions for Further Exploration:

1. How can you find the lengths of $AC$ and $BC$ if given more information about other angles or sides?
2. What properties of isosceles right triangles help in this solution?
3. How would the solution change if the triangle were not isosceles?
4. What would happen if the length of $AE$ were changed to a different value?
5. How do you apply the Pythagorean theorem to solve for unknown sides in right triangles?

Tip:

For isosceles right triangles, remember that the two legs are equal, and the hypotenuse is $\text{leg} \times \sqrt{2}$. This relationship often simplifies problems.