To determine what information is needed to prove that lines $r$ and $s$ are parallel, let's analyze the relationships between angles formed by the transversal $k$ that intersects both lines $r$ and $s$.

Step-by-Step Analysis

1. Identify Angle Pairs for Parallel Line Proofs: According to the parallel line postulates, if a pair of corresponding angles, alternate interior angles, or alternate exterior angles are congruent, then the lines intersected by the transversal are parallel.

2. Evaluate the Angle Options:

   • Option 1: $\angle 5 \cong \angle 6$
     This pair does not help in proving $r \parallel s$ since these are consecutive interior angles on the same side of the transversal. For parallel lines, consecutive interior angles would be supplementary, not necessarily congruent.

   • Option 2: $\angle 4 \cong \angle 5$
     This is not helpful because $\angle 4$ and $\angle 5$ are corresponding angles related to the transversal, but they are on opposite lines without directly forming an angle relationship for $r \parallel s$.

   • Option 3: $\angle 4 \cong \angle 6$
     This pair represents alternate interior angles. If $\angle 4$ and $\angle 6$ are congruent, it would imply that $r \parallel s$ by the Alternate Interior Angle Theorem.

   • Option 4: $\angle 1 \cong \angle 3$
     $\angle 1$ and $\angle 3$ are corresponding angles formed by the transversal. If they are congruent, it would imply $r \parallel s$ by the Corresponding Angles Postulate.

Conclusion

The correct answer is:

• $\angle 4 \cong \angle 6$ because they are alternate interior angles, and proving them congruent would confirm that $r \parallel s$.

Let me know if you'd like further details on the angle relationships or have any questions!

Related Questions

1. What other angle pairs could we use to prove two lines are parallel?
2. How does the Corresponding Angles Postulate help in proving parallel lines?
3. What is the difference between alternate interior and alternate exterior angles?
4. Why is congruency of consecutive interior angles not used to prove parallel lines?
5. Can supplementary angles ever be used to prove lines are parallel?

Tip

Remember, when dealing with parallel lines and a transversal, knowing the types of angle relationships (corresponding, alternate interior, alternate exterior) helps quickly identify congruency conditions.