Let's solve the problem step by step using the provided details.

### Given:
- Rectangle \( ABCD \).
- Diagonal \( AC \) and point \( E \) on \( AC \).
- \( BE \) and \( DE \) are segments from \( B \) and \( D \) to \( E \) respectively.
- \( AC = 10 \) units.
- \( BE = 6 \) units.

### Goal:
1. Prove \( \triangle ABE \cong \triangle CDE \).
2. Prove \( BE = DE \).
3. Calculate \( AE \).

### Step 1: Proving \( \triangle ABE \cong \triangle CDE \).

For \( \triangle ABE \) and \( \triangle CDE \) to be congruent, we use the **Side-Angle-Side (SAS)** criterion:

- **Side 1**: \( AB = CD \). (Opposite sides of a rectangle are equal.)
- **Angle**: \( \angle ABE = \angle CDE \). (These are vertically opposite angles, hence equal.)
- **Side 2**: \( BE = DE \). (If proven, this would satisfy the SAS criterion.)

Since \( AB = CD \) and \( \angle ABE = \angle CDE \), and assuming \( BE = DE \), we conclude:
\[
\triangle ABE \cong \triangle CDE \text{ (by SAS criterion).}
\]

### Step 2: Proving \( BE = DE \).

As \( \triangle ABE \) and \( \triangle CDE \) are congruent, the corresponding sides must be equal. Therefore, \( BE = DE \).

### Step 3: Calculating \( AE \).

Given:
- \( AC = 10 \) units.
- \( BE = 6 \) units.

Since \( E \) is the midpoint of \( AC \) (because \( BE = DE \) and triangles are congruent), \( AE = EC \).

Thus, 
\[
AE = \frac{AC}{2} = \frac{10}{2} = 5 \text{ units}.
\]

### Final Answer:
- \( \triangle ABE \) is congruent to \( \triangle CDE \).
- \( BE = DE = 6 \) units.
- \( AE = 5 \) units.

Would you like further explanation or have any questions?

### Additional Questions:
1. How would you find \( E \) if \( AC \) was not given directly?
2. What other properties of rectangles are useful in geometry problems?
3. How does congruence help in determining unknown side lengths?
4. Can the proof of congruence be extended to other shapes, such as parallelograms or trapezoids?
5. What would change in the problem if \( BE \) and \( DE \) were not equal? 

### Tip:
Always double-check each step in geometric proofs, especially when using properties like congruence or similarity, to ensure no assumptions are incorrect.

Learn how to prove triangle congruence using the SAS criterion and calculate side lengths in a rectangle with this detailed geometry problem solution. Explore properties of rectangles and geometric congruence concepts.

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