https://file.aiquickdraw.com/mathbot/file/172800096753873xcqt8y.jpg

From the image provided, we have a geometric proof involving triangle properties. Let's analyze the given information:

### Given Information:
1. **Step 1:** \( BD \) bisects \( \triangle ABC \). 
   - **Reason:** Given.
   - This indicates that segment \( BD \) is a bisector of the triangle \( \triangle ABC \), meaning it divides the angle \( \triangle ABC \) into two equal parts.

2. **Step 2:** \( AB \cong BC \).
   - **Reason:** Given.
   - This means that side \( AB \) is congruent to side \( BC \). Therefore, \( \triangle ABC \) is isosceles with the two equal sides \( AB \) and \( BC \).

3. **Step 3:** \( BD \cong BD \).
   - **Reason:** Reflexive Property.
   - By the reflexive property, segment \( BD \) is congruent to itself.

### Determining the Next Step in the Proof:
The next logical step is to establish the congruence of triangles to prove something about the entire figure. Based on the information, let's determine the correct statement:

Since \( BD \) is a bisector, and we know \( AB \cong BC \), and \( BD \cong BD \) by the Reflexive Property, we can use **Side-Angle-Side (SAS) Congruence** to conclude that:

### Step 4:
- **Statement:** \( \triangle ABD \cong \triangle CBD \).
- **Reason:** SAS Congruence.
  
This is because:
- **Side:** \( AB \cong BC \) (Given)
- **Angle:** The angle \( \angle ABD \) is congruent to \( \angle CBD \) (since \( BD \) bisects \( \triangle ABC \)).
- **Side:** \( BD \cong BD \) (Reflexive Property).

### Summary:
The correct next statement should be:

- **Statement:** \( \triangle ABD \cong \triangle CBD \)
- **Reason:** SAS Congruence

This statement helps to conclude that the two triangles are congruent by the Side-Angle-Side congruence postulate.

Would you like more details or have any other questions about this proof?

Here are 5 related questions you might be interested in:
1. What is the importance of congruence in geometric proofs?
2. How does the reflexive property help in proving congruence?
3. What are other methods to prove triangle congruence besides SAS?
4. How can you determine if two angles are congruent in an isosceles triangle?
5. Why is the bisector of a triangle significant in geometric constructions?

**Tip:** When working on geometric proofs, always try to connect given information with congruence properties like SAS, ASA, or SSS to establish relationships between different parts of the figure.

The query is about proving congruence in triangle ABC where BD bisects ∠ABC, and AB ≅ BC.

This proof demonstrates triangle congruence in an isosceles triangle using the Side-Angle-Side (SAS) postulate. The problem involves proving congruence between triangles ABD and CBD in a bisected isosceles triangle. Suitable for students in Grades 8-10.

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