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Given: CD//BE; angle 3 is congruent to angle 1. Prove BE bisects angle DBA

Let's analyze the problem using the diagram and the given information. Here’s a step-by-step outline of how to prove that \( BE \) bisects \( \angle DBA \).

### Given:
- \( CD \parallel BE \)
- \( \angle 3 \cong \angle 1 \) (Angle 3 is congruent to Angle 1)

### To prove:
- \( BE \) bisects \( \angle DBA \)

### Proof:

1. **Identify corresponding angles due to parallel lines**:  
   Since \( CD \parallel BE \), we can apply the **Corresponding Angles Theorem**. This tells us that:
   \[
   \angle 1 \cong \angle 2
   \]
   because \( CD \parallel BE \) and the transversal line \( BD \) forms these angles.

2. **Given congruence**:  
   From the problem, we know that:
   \[
   \angle 1 \cong \angle 3
   \]
   (This is given).

3. **Conclude congruence of angles**:  
   From steps 1 and 2, we have:
   \[
   \angle 2 \cong \angle 3
   \]
   (By transitivity of congruence).

4. **Apply Angle Bisector Definition**:  
   Since \( \angle 2 \cong \angle 3 \), and \( BE \) divides \( \angle DBA \) into two congruent angles, it follows that \( BE \) bisects \( \angle DBA \).

Thus, \( BE \) bisects \( \angle DBA \), as required.

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Would you like more details on any of these steps? Here are 5 related questions to further explore this proof:

1. What is the Corresponding Angles Theorem, and how is it used in geometry?
2. How do parallel lines influence angle relationships in a triangle?
3. What is the transitive property of congruence, and how does it apply here?
4. What are the properties of an angle bisector, and how can they be verified?
5. Could this proof be extended to other types of triangles, or does it rely on specific conditions?

### Tip:
When dealing with parallel lines, always look for corresponding, alternate interior, or consecutive angles, as they often lead to key insights in geometric proofs.

Given: CD // BE; angle 3 is congruent to angle 1. Prove BE bisects angle DBA.

In this geometric proof, we explore how to prove that BE bisects angle DBA using the given information CD // BE and congruent angles. We apply the Corresponding Angles Theorem and the Transitive Property of Congruence. Suitable for students in Grades 8-10.

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