https://file.aiquickdraw.com/mathbot/file/1729752056640uut9lee3.jpg

Let's analyze the proof step by step to fill in the missing statement and reason.

The problem is asking to prove that \( \triangle ABC \) is an isosceles triangle, given that \( AC \perp BD \) and \( \triangle ABD \cong \triangle CBD \).

### Given:
- \( AC \perp BD \)
- \( \triangle ABD \cong \triangle CBD \)

### Proof:
We want to show \( \triangle ABC \) is isosceles, which means that two sides of the triangle are congruent.

Let’s review each step:

1. **Step 1 (Given):**
   - **Statement:** \( AC \perp BD \) and \( \triangle ABD \cong \triangle CBD \)
   - **Reason:** Given

2. **Step 2:**
   - **Statement:** \( \angle ADB \) is a right angle.
   - **Reason:** Perpendicular lines form right angles.

3. **Step 3:**
   - **Statement:** \( \angle CDB \) is a right angle.
   - **Reason:** Perpendicular lines form right angles.

4. **Step 4:**
   - **Statement:** \( \angle ADB \cong \angle CDB \)
   - **Reason:** All right angles are congruent.

Now, the missing statement and reason at **Step 5**:
   
   **Missing Step 5:**
   - **Statement:** \( BD \cong BD \)
   - **Reason:** Reflexive property of congruence (a segment is congruent to itself).

6. **Step 6:**
   - **Statement:** \( \triangle ABD \cong \triangle CBD \)
   - **Reason:** ASA (Angle-Side-Angle postulate)

7. **Step 7:**
   - **Statement:** \( AB \cong BC \)
   - **Reason:** Corresponding parts of congruent triangles are congruent (CPCTC).

8. **Step 8:**
   - **Statement:** \( \triangle ABC \) is an isosceles triangle.
   - **Reason:** The triangle has two congruent sides.

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This completes the proof. If you have any further questions, let me know!

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### 5 Related Questions:
1. Why does the reflexive property apply to line segment \( BD \)?
2. How does the ASA postulate work in triangle congruence?
3. What are some other ways to prove that a triangle is isosceles?
4. How can CPCTC be used in other types of geometric proofs?
5. Why do perpendicular lines always form right angles?

### Tip:
Always identify common sides or angles in overlapping triangles to apply properties like the reflexive property for congruence.

Given: AC ⊥ BD and ∠ABD ≅ ∠CBD. Prove: △ABC is an isosceles triangle.

Learn how to prove that △ABC is an isosceles triangle given AC ⊥ BD and ∠ABD ≅ ∠CBD. This proof involves key geometric concepts such as the ASA postulate and CPCTC, and it is suitable for high school students (Grades 9-12).

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