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The two images depict a geometric problem where line segment \( BD \) is the angle bisector of \( \angle ABC \) in a triangle. The task is to complete the flowchart proof that proves \( \triangle ABD \cong \triangle CBD \).

### Steps to complete the flowchart proof:

1. **Given: \( BD \) bisects \( \angle ABC \)**
   - This is already provided as a given.

2. **\( \angle ABD \cong \angle CBD \)**
   - **Reason**: Definition of an angle bisector. Since \( BD \) bisects \( \angle ABC \), the angles on either side of the bisector are congruent.

3. **\( AB = CB \)**
   - **Reason**: This is likely based on the assumption that triangle \( ABC \) is isosceles, which is a common scenario when dealing with angle bisectors in symmetric triangles. Therefore, the sides opposite the congruent angles, \( AB \) and \( BC \), are congruent.

4. **\( BD = BD \)**
   - **Reason**: Reflexive property of equality. A segment is always congruent to itself.

5. **\( \triangle ABD \cong \triangle CBD \)**
   - **Reason**: Angle-Side-Angle (ASA) Congruence Postulate. Two triangles are congruent if two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle.

### Completed Flowchart:
- **Top box**: \( BD \) bisects \( \angle ABC \) - Reason: Given.
- **Next row**: \( \angle ABD \cong \angle CBD \) - Reason: Definition of angle bisector; \( AB = BC \) - Reason: Isosceles triangle or symmetry; \( BD = BD \) - Reason: Reflexive property.
- **Final box**: \( \triangle ABD \cong \triangle CBD \) - Reason: ASA Congruence Postulate.

Would you like further clarification or details on this proof?

### Related Questions:
1. How does the definition of an angle bisector help in proving congruent triangles?
2. What are other methods for proving triangle congruence apart from ASA?
3. How can we prove that triangle \( ABC \) is isosceles if it wasn't given explicitly?
4. What other properties of isosceles triangles can be derived from this configuration?
5. How does the reflexive property help in geometric proofs?

### Tip:
In geometric proofs, always identify congruent sides and angles using known theorems or definitions, such as the reflexive property, angle bisectors, or properties of specific triangles like isosceles triangles.

Given BD is the angle bisector of ∠ABC, complete the flowchart proof that proves ΔABD ≅ ΔCBD.

Learn how to prove that ΔABD ≅ ΔCBD using the angle bisector BD of ∠ABC. This problem covers key geometric concepts like the ASA postulate, reflexive property, and the definition of an angle bisector. Suitable for students in Grades 8-10.

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