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To solve this geometric proof, we need to show that \( \overline{AB} \cong \overline{AC} \) (that is, the line segments are congruent), given that both are tangents to two different circles. Here's how we can organize the proof:

### Given:
- \( \overline{AB} \) and \( \overline{AD} \) are tangent to circle \( X \).
- \( \overline{AC} \) and \( \overline{AD} \) are tangent to circle \( Y \).

### To Prove:
- \( \overline{AB} \cong \overline{AC} \).

### Hint:
- We will use the fact that tangents from a common external point to a circle are congruent.

### Proof Outline:

#### Statements and Reasons:

1. **Statement**: \( \overline{AB} \) and \( \overline{AD} \) are tangents from point \( A \) to circle \( X \).  
   **Reason**: Given.

2. **Statement**: \( \overline{AB} \cong \overline{AD} \).  
   **Reason**: Tangents from the same external point to a circle are congruent.

3. **Statement**: \( \overline{AC} \) and \( \overline{AD} \) are tangents from point \( A \) to circle \( Y \).  
   **Reason**: Given.

4. **Statement**: \( \overline{AC} \cong \overline{AD} \).  
   **Reason**: Tangents from the same external point to a circle are congruent.

5. **Statement**: \( \overline{AB} \cong \overline{AC} \).  
   **Reason**: Transitive property of congruence (if \( \overline{AB} \cong \overline{AD} \) and \( \overline{AC} \cong \overline{AD} \), then \( \overline{AB} \cong \overline{AC} \)).

Thus, we have proven that \( \overline{AB} \cong \overline{AC} \), completing the proof.

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Let me know if you'd like further clarification or if you have any questions!

Here are some related questions:
1. What is the transitive property of congruence?
2. How do tangents from the same external point relate to congruence?
3. What are the key properties of tangents to a circle?
4. How would the proof change if the tangents were from different external points?
5. Can this concept be applied to non-circular shapes?

**Tip:** Always remember that tangents from a common external point to the same circle are congruent—a key fact often used in circle-related proofs.

Given: \( \overline{AB} \) and \( \overline{AD} \) are tangent to \( \odot X \); \( \overline{AC} \) and \( \overline{AD} \) are tangent to \( \odot Y \). Prove: \( \overline{AB} \cong \overline{AC} \). Hint: Prove tangents congruent.

Learn how to prove that two tangents drawn from an external point to different circles are congruent. This geometry problem uses the theorem that tangents from the same external point to a circle are congruent, suitable for students in Grades 9-12.

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