https://file.aiquickdraw.com/mathbot/file/17269768564289hwb32ce.jpg

Let's complete the proof shown in the image step-by-step.

### Given:
1. \( \overline{AB} \) and \( \overline{AC} \) are tangent to circle \( O \).
2. \( \overline{BD} \equiv \overline{CD} \).

### To Prove:
\( \overline{BD} \equiv \overline{CD} \).

### Proof Structure:
1. **Statements**: The sequential steps to prove the theorem.
2. **Reasons**: The justification for each statement.

### Proof Completion:
1. **Statement 1**: \( \overline{AB} \) and \( \overline{AC} \) are tangent to circle \( O \).
   - **Reason 1**: Given.

2. **Statement 2**: \( \overline{OA} \equiv \overline{OA} \).
   - **Reason 2**: Reflexive Property.

3. **Statement 3**: \( \angle OAB \equiv \angle OAC \).
   - **Reason 3**: Tangents to a circle from a common external point are equal, hence \( \angle OAB \) and \( \angle OAC \) are congruent.

4. **Statement 4**: \( \triangle OAB \equiv \triangle OAC \).
   - **Reason 4**: SAS Postulate (Side-Angle-Side).

5. **Statement 5**: \( \overline{BD} \equiv \overline{CD} \).
   - **Reason 5**: Corresponding parts of congruent triangles are congruent (CPCTC).

6. **Statement 6**: \( \angle OBD \equiv \angle OCD \).
   - **Reason 6**: CPCTC.

7. **Statement 7**: \( \overline{BD} \equiv \overline{CD} \).
   - **Reason 7**: CPCTC.

This completes the proof, showing that \( \overline{BD} \equiv \overline{CD} \).

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Do you want any additional details or have any questions? Here are 5 related questions you might find useful:

1. What is the significance of the SAS Postulate in geometric proofs?
2. How do tangents drawn from the same external point relate to each other?
3. What does CPCTC stand for, and why is it important in congruence proofs?
4. How can you prove that two segments are congruent without using triangle congruence?
5. Why is it necessary to prove triangle congruence in this problem?

**Tip**: In geometric proofs, always try to identify congruent triangles as they can help establish the congruence of other elements like angles and segments.

Given that AB and AC are tangent to circle O, prove that BD is congruent to CD.

This problem involves proving that two segments are congruent using tangent properties and triangle congruence. We explore key theorems such as the SAS Postulate and CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to establish the solution. Suitable for high school students in Grades 9-11.

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