https://file.aiquickdraw.com/mathbot/file/1726542657830ow4e4h1w.jpg

To complete the flowchart proof based on the given triangle diagram, let's analyze the situation:

1. **Given Information:**
   - \( \triangle ACB \) is an isosceles triangle because sides \( AB \) and \( CB \) are marked as congruent.
   - Segment \( AD \) is congruent to segment \( CD \).
   - Segment \( BD \) is common to both triangles.

The proof seems to be about showing that \( \triangle ABD \cong \triangle CBD \).

2. **Flowchart Proof Explanation:**
   - **Step 1:** \( AD \cong CD \) (Given in the diagram).
     - **Reason:** Definition of midpoint or segment congruence.
   - **Step 2:** \( AB \cong CB \) (Given in the diagram).
     - **Reason:** Given that \( \triangle ACB \) is isosceles.
   - **Step 3:** \( BD \cong BD \) (This is a shared side between the two triangles).
     - **Reason:** Reflexive property of congruence.
   - **Step 4:** \( \triangle ABD \cong \triangle CBD \).
     - **Reason:** Side-Side-Side (SSS) Congruence Postulate.

### Detailed Flowchart for the Proof:
1. **AD ≅ CD**
   - **Reason:** Given (or midpoint if D is the midpoint of AC).
   
2. **AB ≅ CB**
   - **Reason:** Given (or property of an isosceles triangle).

3. **BD ≅ BD**
   - **Reason:** Reflexive property of congruence (a side is congruent to itself).

4. **ΔABD ≅ ΔCBD**
   - **Reason:** Side-Side-Side (SSS) Congruence Postulate.

Let me know if you need more details on any part or have further questions.

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Here are five questions related to the proof:

1. What other methods besides SSS could be used to prove congruence between two triangles?
2. How does the reflexive property help in triangle congruence proofs?
3. What are the key characteristics of an isosceles triangle?
4. Can the angle-side-angle (ASA) postulate be used in this proof?
5. Why is it important to specify reasons in a flowchart proof?

**Tip:** When proving triangles congruent, always check if they share common sides or angles—it often helps simplify the proof.

Based on the given diagram, complete the flowchart proof below. The diagram shows an isosceles triangle with segments AD ≅ CD, AB ≅ CB, and BD ≅ BD. Prove that triangles ABD and CBD are congruent.

Explore the flowchart proof for proving that triangles ABD and CBD are congruent using the Side-Side-Side (SSS) Congruence Postulate. The problem involves an isosceles triangle with given congruent sides. Suitable for students in Grades 8-10.

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