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This question is asking you to complete the flowchart proof for two triangles, \( \triangle ABD \) and \( \triangle CBD \), with the given that \( D \) is the midpoint of \( AC \). Let's walk through the reasoning behind each step:

### Given:
- \( D \) is the midpoint of \( AC \).

### Step 1: \( AD \cong CD \) 
- **Reason**: Since \( D \) is the midpoint of \( AC \), it divides \( AC \) into two equal segments, so \( AD = CD \).
- **Reason**: **Definition of midpoint**

### Step 2: \( \angle A \cong \angle C \)
- **Reason**: \( \triangle ABC \) is isosceles (as indicated by the tick marks on \( AB \) and \( BC \), showing that \( AB \cong BC \)), so the base angles \( \angle A \) and \( \angle C \) are congruent.
- **Reason**: **Base angles of an isosceles triangle are congruent**

### Step 3: \( AB \cong CB \)
- **Reason**: This is given directly from the diagram as the two sides \( AB \) and \( BC \) have the same number of tick marks, indicating they are congruent.
- **Reason**: **Given**

### Step 4: \( \triangle ABD \cong \triangle CBD \)
- **Reason**: At this point, we have two congruent sides \( AB \cong CB \), two congruent angles \( \angle A \cong \angle C \), and \( AD \cong CD \). This is enough to conclude that \( \triangle ABD \cong \triangle CBD \) by the **SAS (Side-Angle-Side) Congruence Postulate**.

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Would you like further clarification on any of the reasoning steps? Here are five related questions to deepen your understanding:

1. What other congruence postulates or theorems could be applied if different information was given in this triangle?
2. Why does the midpoint ensure that \( AD = CD \)?
3. What does the isosceles triangle theorem state, and how does it apply here?
4. How can you prove two triangles are congruent using the Side-Angle-Side (SAS) postulate?
5. What role do base angles play in determining the congruence of triangles?

**Tip**: When proving triangles congruent, always look for congruent sides and angles based on givens, definitions, and postulates like SAS, SSS, or ASA.

Given D is the midpoint of AC, complete the flowchart proof.

Solve a flowchart proof to show triangle congruence in an isosceles triangle where D is the midpoint of AC. Use the Midpoint Theorem and SAS Congruence Postulate to prove ΔABD ≅ ΔCBD. Suitable for students in Grades 9-10.

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