Question

given: lines a and b are parallel and line c is a transversal.
prove: ∠2 is supplementary to ∠8
what is the missing reason in the proof?

1. a || b, c is a transversal (given)
2. ∠6 ≅ ∠2 (?)
3. m∠6 = m∠2 (def. of congruent)
4. ∠6 is supp. to ∠8 (def. of linear pair)
5. ∠2 is supp. to ∠8 (congruent supplements theorem)

corresponding angles theorem
alternate interior angles theorem
vertical angles theorem
alternate exterior angles theorem

Explanation:

• Corresponding angles theorem: Applies to angles in the same relative position. $\angle6$ and $\angle2$ are not in corresponding positions.
• Alternate interior angles theorem: Applies to angles between the parallel lines and on opposite sides of the transversal. $\angle6$ and $\angle2$ are not alternate - interior.
• Vertical angles theorem: Applies to angles opposite each other when two lines intersect. Not relevant here.
• Alternate exterior angles theorem: When two parallel lines are cut by a transversal, alternate exterior angles are congruent. $\angle6$ and $\angle2$ are alternate exterior angles (outside the parallel lines \(a\) and \(b\) and on opposite sides of transversal \(c\)).

Answer:

alternate exterior angles theorem