In the given image, we are tasked with determining which pairs of lines are parallel and justifying the answer using a specific theorem. The diagram shows several intersecting lines, and we are presented with six possible answers, each involving a different pair of lines and the application of various theorems.

Let's analyze this:

• The diagram indicates two sets of transversal lines intersecting two potential pairs of parallel lines.
• We can use the Converse of the Alternate Interior Angles Theorem or Converse of the Corresponding Angles Theorem to determine which pairs of lines are parallel.

Answer justification:

Looking at the arrangement, lines $K \parallel C$ and $H \parallel N$ are transversals. Given the position of the angles, we can use the Converse of the Corresponding Angles Theorem to deduce that:

• Line $KH \parallel CN$.

Thus, the correct answer is option E, as it states:

• $KC \parallel HN$ by the Converse of the Corresponding Angles Theorem.

Would you like further details or clarification on this?

Here are 5 related questions to explore this topic further:

1. What is the Converse of the Alternate Interior Angles Theorem?
2. How do corresponding angles prove parallelism in a transversal diagram?
3. What is the difference between alternate interior and alternate exterior angles?
4. Can same-side interior angles be used to prove parallel lines?
5. How can you identify transversal lines in a complex figure?

Tip: When working with parallel lines and a transversal, identifying angle relationships (corresponding, alternate, or same-side) is key to proving which lines are parallel.