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To complete the proof that \(AB = BE\) using the given statements and diagram, we should follow the plan of proving triangles congruent and then using congruent parts of congruent triangles are congruent (CPCTC).

Given:
- \(BD\) bisects \(\angle CBE\)
- \(AE \parallel BD\)

**Proof:**

1. **Statement:** \(BD\) bisects \(\angle CBE\).
   **Reason:** Given.

2. **Statement:** \(\angle EBD = \angle CBD\).
   **Reason:** Definition of angle bisector (from statement 1).

3. **Statement:** \(\angle AEB = \angle EBD\).
   **Reason:** Corresponding angles are congruent since \(AE \parallel BD\) and \(BE\) is a transversal (hint from "If a transversal intersects two parallel lines, then the corresponding angles are congruent").

4. **Statement:** \(\triangle AEB \cong \triangle BEB\).
   **Reason:** Angle-Angle-Side (AAS) Congruence (from statements 2 and 3, and \(BE\) is common to both triangles).

5. **Statement:** \(AB = BE\).
   **Reason:** CPCTC (Congruent Parts of Congruent Triangles are Congruent), using the congruence from statement 4.

6. **Conclusion:** \(AB = BE\).
   **Reason:** Proven above.

This completes the geometric proof, showing that the segment \(AB\) is indeed equal to \(BE\), given that \(BD\) bisects \(\angle CBE\) and \(AE \parallel BD\).

Would you like more detail on any part of this proof or have other questions? Here are a few related topics you might find interesting:

1. What is the Angle-Angle-Side (AAS) theorem and how is it used in proofs?
2. How does the definition of an angle bisector help in solving problems involving triangles?
3. Can you explain why corresponding angles are congruent when a transversal intersects parallel lines?
4. What are other instances where CPCTC could be applied in geometric proofs?
5. How does the property of parallel lines create equal angles in geometric figures?

**Tip:** Always remember to include all known information in your diagram when solving geometry proofs. It helps visualize the problem and identify congruent parts and angles more easily.

Prove that AB ≅ BE given BD bisects ∠CBE and AE ∥ BD.

Complete a geometric proof to show AB ≅ BE using triangle congruence and properties of parallel lines and bisectors. This solution explores the Angle Bisector Theorem, Angle-Angle-Side (AAS) Congruence, and CPCTC, suitable for students in Grades 9-10.

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