The relationship that proves lines f and g are parallel is B. ∠1 ≅ ∠5. This is the corresponding angles postulate.
Corresponding angles are two angles formed when a transversal intersects two parallel lines, and they are located on the same side of the transversal and outside the parallel lines.
In the diagram, ∠1 and ∠5 are corresponding angles.
The corresponding angles postulate states that if two lines are cut by a transversal, and the corresponding angles are congruent, then the lines are parallel.
Here's why the other options are not correct:
A. ∠1 ≅ ∠4:
These are alternate interior angles, but the converse of the alternate interior angles theorem is not true.
Just because alternate interior angles are congruent does not mean the lines are parallel.
C. ∠1 ≅ ∠6:
These are alternate exterior angles, but the converse of the alternate exterior angles theorem is not true either.
Just because alternate exterior angles are congruent does not mean the lines are parallel.
D. ∠1 ≅ ∠7:
These are same-side interior angles, but for lines to be parallel, the same-side interior angles must be supplementary (add up to 180 degrees).
In this case, we don't have enough information to determine whether ∠1 and ∠7 are supplementary.
The relationship that proves lines f and g are parallel is B. ∠1 ≅ ∠5.
This is the corresponding angles postulate.
Therefore, based on the corresponding angles postulate, the only relationship that proves lines f and g are parallel is B. ∠1 ≅ ∠5.