https://file.aiquickdraw.com/mathbot/file/17285325087369dpnkj9q.jpg

Based on the image, the question asks which lines are parallel if ∠4 and ∠6 are congruent. To determine this, let's analyze the setup:

The diagram in the image shows a set of lines with transversal lines crossing them, which typically form angles labeled in a specific pattern. If ∠4 and ∠6 are congruent, this usually indicates a specific relationship due to the properties of corresponding angles, alternate interior angles, or other angle relationships formed by a transversal intersecting parallel lines.

### Analysis:
- If ∠4 and ∠6 are alternate interior angles and they are congruent, then by the Alternate Interior Angles Theorem, the lines they lie between are parallel.

Based on this reasoning, the correct answer choice depends on how the lines are labeled in the diagram (a, b, c, etc.). If ∠4 and ∠6 are alternate interior angles, the lines that they lie between would be the parallel lines.

Given the choices:
1. \( a \parallel b \)
2. \( b \parallel c \)
3. \( a \parallel c \)

The correct answer would be the pair that aligns with the lines ∠4 and ∠6 are between.

If you have any further questions or need more details, let me know! 

### Follow-up Questions:
1. What is the role of alternate interior angles in determining parallel lines?
2. How does the Congruent Angles Theorem apply to parallel lines?
3. Could you explain the difference between corresponding angles and alternate interior angles?
4. What other angle relationships can indicate parallel lines?
5. Why is it important for the lines to be straight when using these angle theorems?

**Tip:** Always double-check the diagram and the labeling of angles when determining relationships like congruence or parallelism.

Use the diagram to answer the two-part question. Part A: Which lines are parallel if ∠4 and ∠6 are congruent?

Learn how to determine parallel lines using angle congruence in geometry. This lesson covers the Alternate Interior Angles Theorem and the Congruent Angles Theorem, suitable for Grade 9 students.

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