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To determine which lines, if any, must be parallel based on the given information, we need to apply the properties of parallel lines and the angles formed when a transversal crosses them.

Given: ∠3 is congruent to ∠13.

In the diagram, angles ∠3 and ∠13 are positioned such that they are alternate interior angles with respect to the lines \(a\) and \(b\) and the transversal \(c\).

By the Converse of the Alternate Interior Angles Theorem, if alternate interior angles are congruent, then the lines are parallel. Therefore, since ∠3 is congruent to ∠13, we can conclude that lines \(a\) and \(b\) must be parallel.

Thus, the correct answer is:

- \( a \parallel b \), Converse of the Alternate Interior Angles Theorem

Would you like more details on this explanation, or do you have any questions?

Here are some more questions you might find interesting:

1. How do you prove lines are parallel using corresponding angles?
2. What is the Converse of the Same-Side Interior Angles Theorem?
3. Can you explain how the Corresponding Angles Postulate works?
4. What is the difference between alternate interior and alternate exterior angles?
5. How do you identify transversals in geometric figures?

**Tip:** Remember that the Converse of the Alternate Interior Angles Theorem is a powerful tool for proving parallel lines when you know two angles are congruent.

Learn how to determine parallel lines using the Converse of the Alternate Interior Angles Theorem with a step-by-step explanation. This problem involves identifying congruent angles and applying geometric properties to conclude the parallelism of lines. Suitable for high school students.

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