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The image shows a geometry proof involving parallel lines, a transversal, and angle relationships. The problem asks for the missing reason in a proof that shows ∠2 is supplementary to ∠8, given that lines \(a\) and \(b\) are parallel, and line \(c\) is a transversal.

### Proof Summary:
- **Statement 1**: \(a \parallel b\), \(c\) is a transversal (Given)
- **Statement 2**: \( \angle 6 \cong \angle 2 \) (Reason: ?)
- **Statement 3**: \( m \angle 6 = m \angle 2 \) (Definition of congruent angles)
- **Statement 4**: \( \angle 6 \) is supplementary to \( \angle 8 \) (Definition of a linear pair)
- **Statement 5**: \( \angle 2 \) is supplementary to \( \angle 8 \) (Congruent Supplements Theorem)

### The missing reason in Statement 2 is the **Alternate Interior Angles Theorem**. 

This theorem states that if two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. In this case, since lines \(a \parallel b\) and \(c\) is the transversal, \( \angle 6 \cong \angle 2 \) by the Alternate Interior Angles Theorem.

Thus, the correct answer to the question is:
- **Alternate interior angles theorem**.

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Would you like further details on any part of this proof? Here are some related questions to explore:

1. Why does the congruent supplements theorem apply in this situation?
2. How does the definition of a linear pair help prove angles are supplementary?
3. What is the role of transversals in geometry involving parallel lines?
4. How would the proof change if lines \(a\) and \(b\) were not parallel?
5. What other angle relationships can be established when a transversal intersects two parallel lines?

**Tip**: Remember that alternate interior angles are congruent when two parallel lines are cut by a transversal.

What is the missing reason in the proof: Given lines a and b are parallel and line c is a transversal, prove ∠2 is supplementary to ∠8?

This geometry problem involves proving that ∠2 is supplementary to ∠8 given that lines a and b are parallel and line c is a transversal. Using the Alternate Interior Angles Theorem, Definition of Linear Pair, and Congruent Supplements Theorem, this proof is ideal for students in grades 8-10.

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