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Let's go through the proof using the given hint: **Use the LL Theorem**, which stands for the **Leg-Leg Theorem** (essentially a version of the Hypotenuse-Leg theorem for right triangles). This theorem is useful for proving the congruence of two right triangles based on the lengths of two corresponding legs.

### Given Information:
- Rectangle \( ABCD \), where \( E \) is the midpoint of \( \overline{AD} \).
- You need to prove that \( \triangle BAE \cong \triangle CDE \).

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Here is a possible flow for the proof:

| Statements                                | Reasons                               |
|-------------------------------------------|---------------------------------------|
| 1. \( ABCD \) is a rectangle              | 1. Given                             |
| 2. \( \overline{AB} \parallel \overline{CD} \), \( \overline{AD} \parallel \overline{BC} \) | 2. Definition of a rectangle (opposite sides are parallel) |
| 3. \( \overline{AE} = \overline{ED} \)    | 3. \( E \) is the midpoint of \( \overline{AD} \) |
| 4. \( \overline{AB} = \overline{CD} \)    | 4. Opposite sides of a rectangle are congruent |
| 5. \( \angle AEB = \angle CED = 90^\circ \) | 5. Definition of a rectangle (right angles) |
| 6. \( \triangle BAE \cong \triangle CDE \) | 6. LL Theorem (Leg-Leg Theorem)       |

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### Explanation:
- **Step 1**: Given information that \( ABCD \) is a rectangle.
- **Step 2**: A rectangle has parallel opposite sides, which is a property of rectangles.
- **Step 3**: Since \( E \) is the midpoint of \( \overline{AD} \), the two segments \( \overline{AE} \) and \( \overline{ED} \) are equal in length.
- **Step 4**: Opposite sides of a rectangle are equal, hence \( \overline{AB} = \overline{CD} \).
- **Step 5**: Both triangles \( \triangle BAE \) and \( \triangle CDE \) contain right angles at \( E \), as all angles in a rectangle are \( 90^\circ \).
- **Step 6**: Using the Leg-Leg Theorem (LL), the two triangles are congruent because they have two pairs of equal corresponding sides (legs) and a right angle.

Would you like more details on any of the steps? 

### Related Questions:
1. How does the Leg-Leg Theorem differ from the Hypotenuse-Leg Theorem?
2. Why are opposite sides of a rectangle always congruent?
3. Can you prove triangle congruence using only angles?
4. What other properties of rectangles help in proving triangle congruence?
5. How does midpoint play a role in geometry proofs like this?

### Tip:
When proving triangle congruence, always identify corresponding sides and angles that can help apply congruence theorems like SAS, SSS, ASA, or LL.

Given: Rectangle ABCD with E the midpoint of AD. Prove: Triangle BAE is congruent to triangle CDE using the LL Theorem.

In this geometry proof, learn how to prove triangle BAE is congruent to triangle CDE using the LL (Leg-Leg) Theorem. This proof leverages properties of rectangles and the midpoint concept. Suitable for students in Grades 8-10.

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