The theorem that we can use to show that ∠1 ≅ ∠7 is: The lines are parallel by the converse of the Alternate Exterior Angles Theorem
How to use the Alternate Exterior angle theorem?
The Alternate Exterior Angles Theorem states that when two parallel lines are cut by a transversal, the resulting alternate exterior angles are congruent. This means that if two parallel lines are intersected by a transversal, then the exterior angles formed on either side of the transversal are equal.
The converse of this theorem is also true: if two lines are cut by a transversal so that the alternate exterior angles are congruent, then the lines are parallel
We are told that:
∠1 ≅ ∠7
This means that they are congruent and looking at the diagram, we see that they are alternate exterior angles