https://file.aiquickdraw.com/mathbot/file/1726981181928p2oy01bn.jpg

Let's go through the proof step-by-step using the given information.

### Given:
- Rectangle \( ABCD \) with \( E \) as the midpoint of \( AD \).
- We are to prove that \( \triangle BAE \cong \triangle CDE \).
- Hint: Use the **LL Theorem** (Leg-Leg Congruence for right triangles).

### Statements and Reasons:
1. \( ABCD \) is a rectangle.  
   **Reason:** Given.

2. \( \overline{AE} \cong \overline{ED} \).  
   **Reason:** \( E \) is the midpoint of \( AD \), definition of midpoint.

3. \( \angle AEB = \angle DEC = 90^\circ \).  
   **Reason:** All angles in a rectangle are \( 90^\circ \), and \( AE \) and \( ED \) are altitudes forming right angles with sides \( AB \) and \( CD \).

4. \( \overline{AB} \cong \overline{CD} \).  
   **Reason:** Opposite sides of a rectangle are congruent.

5. \( \triangle BAE \cong \triangle CDE \).  
   **Reason:** By the **LL Theorem**, since we have two congruent legs (\( AE = ED \) and \( AB = CD \)) and the right angles at \( E \).

6. Therefore, \( \triangle BAE \cong \triangle CDE \).  
   **Reason:** Proven by the LL theorem.

---

### Final Proof Outline:

| **Statements**                        | **Reasons**                              |
|---------------------------------------|------------------------------------------|
| 1. \( ABCD \) is a rectangle.         | Given.                                   |
| 2. \( \overline{AE} \cong \overline{ED} \) | \( E \) is the midpoint of \( AD \).       |
| 3. \( \angle AEB = \angle DEC = 90^\circ \) | All angles in a rectangle are \( 90^\circ \). |
| 4. \( \overline{AB} \cong \overline{CD} \) | Opposite sides of a rectangle are congruent. |
| 5. \( \triangle BAE \cong \triangle CDE \) | LL Theorem. |
| 6. Therefore, \( \triangle BAE \cong \triangle CDE \). | Proven by LL Theorem. |


Let me know if you need any further details or clarifications. Here are 5 related questions for further practice:

1. How does the LL Theorem differ from the SAS theorem for proving triangle congruence?
2. Why are all angles of a rectangle always right angles?
3. What other theorems can be used to prove triangle congruence apart from the LL Theorem?
4. How does the midpoint of a line segment help in proving triangle congruence?
5. Can the LL Theorem be applied to non-right triangles?

**Tip:** In geometry, always look for congruent parts of triangles when attempting to prove them congruent, focusing on corresponding sides and angles.

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

Learn how to prove that triangle BAE is congruent to triangle CDE using the LL theorem. This problem involves rectangle properties and midpoint definitions, making it suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation