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.

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.