To complete the proof that $j \parallel n$ by matching statements with their corresponding reasons, we need to evaluate each statement provided and pair it with the correct justification based on the properties of parallel lines and angles.

Here's how we can logically approach the matching:

1. Statement: "j parallel to k and angle 2 + angle 3 = 180 degrees"

   • Reason: This is Given as per the information in the problem setup.

2. Statement: "angle 1 + angle 3 = 180 degrees"

   • Reason: This follows from the fact that Angles with the same measure are congruent to each other. Since $j \parallel k$, and by the given, $\angle 2 + \angle 3 = 180^\circ$. Since angle 1 is corresponding to angle 2, they are congruent.

3. Statement: "angle 1 + angle 3 = angle 2 + angle 3"

   • Reason: This is based on the fact that Angles with the same measure are congruent to each other.

The final matches should look like this:

• "j parallel to k and angle 2 + angle 3 = 180 degrees" → Given
• "angle 1 + angle 3 = 180 degrees" → Angles with the same measure are congruent to each other
• "angle 1 + angle 3 = angle 2 + angle 3" → Angles with the same measure are congruent to each other

Would you like more details on the concepts used in this proof?

Here are some questions for further exploration:

1. Why does the sum of angle 2 and angle 3 being 180 degrees imply that $j \parallel k$?
2. How can the concept of corresponding angles be used in parallel line proofs?
3. What other angle pairs are formed by a transversal intersecting parallel lines?
4. Can this reasoning apply to non-parallel lines as well?
5. How does the congruence of angles support the establishment of parallel lines?

Tip: Always check for corresponding and alternate interior angles when working with parallel lines and transversals.