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To help you complete this proof, let’s analyze the statements and reasons provided. Based on the information and the given reasons, I’ll determine the missing statements:

### Given Information and Observations:
1. \( \angle BAD \cong \angle BCD \)
2. \( \angle BDA \) and \( \angle BDC \) are right angles.
3. \( \overline{AB} \cong \overline{BC} \)
4. \( \overline{BD} \) bisects \( \angle ABC \).
5. \( \overline{BD} \perp \overline{AC} \), and \( \triangle ABC \) is isosceles with base \( \overline{AC} \).

### Proof Breakdown:

| Statements | Reasons |
|------------|---------|
| 1. \( \overline{BD} \perp \overline{AC} \), \( \triangle ABC \) is isosceles with base \( \overline{AC} \) | 1. Given |
| 2. \( \angle BDA \) and \( \angle BDC \) are right angles | 2. Definition of perpendicular lines |
| 3. \( \angle BDA \cong \angle BDC \) | 3. All right angles are congruent |
| 4. \( \overline{AB} \cong \overline{BC} \) | 4. Definition of isosceles triangle |
| 5. \( \angle BAD \cong \angle BCD \) | 5. Isosceles Triangle Theorem |
| 6. \( \triangle BAD \cong \triangle BCD \) | 6. AAS (Angle-Angle-Side) |
| 7. \( \angle ABD \cong \angle CBD \) | 7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |
| 8. \( \overline{BD} \) bisects \( \angle ABC \) | 8. Definition of angle bisector |

### Explanation:
1. **Statement 1:** Restates the given information regarding perpendicularity and the isosceles property of \( \triangle ABC \).
2. **Statement 2:** Establishes that \( \angle BDA \) and \( \angle BDC \) are right angles based on the perpendicularity.
3. **Statement 3:** Confirms that all right angles are congruent, making \( \angle BDA \cong \angle BDC \).
4. **Statement 4:** Uses the definition of an isosceles triangle to conclude that \( \overline{AB} \cong \overline{BC} \).
5. **Statement 5:** Uses the Isosceles Triangle Theorem to establish \( \angle BAD \cong \angle BCD \).
6. **Statement 6:** Applies the AAS (Angle-Angle-Side) congruence criterion to show that \( \triangle BAD \cong \triangle BCD \).
7. **Statement 7:** Concludes that \( \angle ABD \cong \angle CBD \) by CPCTC, as they are corresponding parts of the congruent triangles.
8. **Statement 8:** Uses the definition of an angle bisector, as \( \overline{BD} \) bisects \( \angle ABC \).

Would you like further explanations on any specific steps?

### Follow-up Questions:

1. What is the significance of the AAS (Angle-Angle-Side) criterion in proving triangle congruence?
2. How does CPCTC (Corresponding Parts of Congruent Triangles are Congruent) apply to this proof?
3. Why does the definition of an isosceles triangle help establish \( \overline{AB} \cong \overline{BC} \)?
4. Can you explain why right angles are considered congruent in all cases?
5. How does the definition of an angle bisector apply in geometrical proofs?

### Tip:
In geometry proofs, it’s essential to rely on definitions, postulates, and theorems systematically to build your reasoning.

Complete the geometry proof involving congruent triangles, right angles, and the isosceles triangle theorem.

Learn how to complete a geometry proof involving congruent triangles, right angles, and the isosceles triangle theorem. This problem covers essential concepts like AAS criterion, CPCTC, and the definition of angle bisectors, ideal for high school students in Grades 9-11.

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