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This problem asks to **prove the Alternate Exterior Angles Converse Theorem**, which states:

"If two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel."

### Given:
- ∠1 ≅ ∠2 (alternate exterior angles)

### Prove:
- \( p \parallel q \) (lines are parallel)

### Proof Outline:

1. **Identify the transversal**: The line cutting through both \( p \) and \( q \) is the transversal (diagonal line in the diagram).
   
2. **Recognize alternate exterior angles**: ∠1 and ∠2 are on opposite sides of the transversal and outside the two lines \( p \) and \( q \), making them alternate exterior angles.

3. **Use the Converse of the Alternate Exterior Angles Theorem**: Since it's given that ∠1 ≅ ∠2, by the Converse of the Alternate Exterior Angles Theorem, the lines \( p \) and \( q \) must be parallel.

4. **Conclusion**: \( p \parallel q \).

Would you like a more detailed step-by-step formal proof or any clarification on this concept?

Here are some related questions to expand understanding:
1. What is the difference between the Alternate Exterior Angles Theorem and its converse?
2. How do we define transversal lines in geometry?
3. What is the significance of proving lines are parallel using angle relationships?
4. Can you apply the Alternate Interior Angles Theorem to this diagram?
5. What other methods could be used to prove two lines are parallel?

**Tip**: Always look for transversal and angle pairs like alternate exterior or alternate interior when working with parallel lines.

Prove Alternate Exterior Angles Converse given that ∠1 ≅ ∠2 and prove that p is parallel to q.

Understand how to prove that two lines are parallel using the Converse of the Alternate Exterior Angles Theorem. This proof involves geometry concepts like transversals and alternate exterior angles. Ideal for students in Grades 8-10.

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