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Add more congruency that builds up to the last 

In this problem, we have two segments \( BD \) and \( AC \) that bisect each other at point \( E \). The goal is to prove that \( AB \parallel CD \). Here’s how we can add more steps with congruency to build up to the conclusion. 

From the current steps:

1. **Given:** \( BD \) and \( AC \) bisect each other.
   - **Reason:** Given.

2. **\( AE \cong EC \) and \( BE \cong ED \)**
   - **Reason:** A segment bisector divides a segment into two congruent segments.

3. **\( \angle AEB \cong \angle CED \)**
   - **Reason:** Vertical angles are congruent.

To expand on this, we can use the concept of congruent triangles to strengthen the proof.

### Additional Steps

4. **\( \triangle AEB \cong \triangle CED \)**
   - **Reason:** Side-Angle-Side (SAS) Congruence Postulate, because \( AE \cong EC \), \( \angle AEB \cong \angle CED \), and \( BE \cong ED \).

5. **\( \angle BAE \cong \angle DCE \) and \( \angle ABE \cong \angle CDE \)**
   - **Reason:** Corresponding parts of congruent triangles are congruent (CPCTC).

6. **\( AB \parallel CD \)**
   - **Reason:** If alternate interior angles are congruent, then the lines are parallel.

### Explanation

By proving that \( \triangle AEB \cong \triangle CED \), we show that the corresponding angles \( \angle BAE \) and \( \angle DCE \), as well as \( \angle ABE \) and \( \angle CDE \), are congruent. These congruent angles create alternate interior angles between lines \( AB \) and \( CD \) when crossed by the transversal \( AC \), thus establishing that \( AB \parallel CD \).

Would you like further details on any of these steps, or have any specific questions?

### Related Questions
1. What is the Side-Angle-Side (SAS) Congruence Postulate and why is it used here?
2. How do alternate interior angles help in proving that two lines are parallel?
3. What is the significance of CPCTC (Corresponding Parts of Congruent Triangles are Congruent) in proofs?
4. How can we use the concept of congruency to prove other parallel lines in different geometric figures?
5. What are some other methods to prove lines are parallel in a quadrilateral?

### Tip
Always look for congruent triangles when you have bisected segments and vertical angles, as they often provide key insights for establishing parallel lines or equal angles.

Add more congruency that builds up to the last.

Explore how to prove AB is parallel to CD by using congruent triangles and the concept of alternate interior angles. This geometry problem involves bisection, vertical angles, and CPCTC to show parallelism between two lines.

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