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To determine which angle must be supplementary to angle 6 to prove that lines \( m \) and \( n \) are parallel, we need to look at the properties of supplementary angles created when a transversal intersects parallel lines.

### Solution:
In the diagram:
- When two lines are parallel, corresponding angles, alternate interior angles, and consecutive interior angles (same-side interior) follow specific rules.
- To prove lines \( m \) and \( n \) are parallel using angle relationships, we look at the consecutive interior angles on the same side of the transversal (line \( t \)).

Here:
- Angle 6 and Angle 3 are consecutive interior angles on the same side of transversal \( t \).
- For lines \( m \) and \( n \) to be parallel, angles 6 and 3 must be supplementary (adding up to \( 180^\circ \)).

Thus, the correct answer is:
**Angle 3.**

Would you like a detailed breakdown of angle relationships in parallel lines or have other questions?

### Related Questions:
1. What are corresponding angles, and how are they used to prove lines are parallel?
2. How do alternate interior angles help in proving lines are parallel?
3. What is the definition of supplementary angles?
4. How do we determine if two angles are supplementary or complementary?
5. What other angle pairs could indicate that lines are parallel?

**Tip:** When analyzing parallel lines and a transversal, focus on identifying corresponding, alternate, and consecutive interior angles for quick verification.

Which of the following must be supplementary to angle 6 to prove lines m and n are parallel?

Learn how supplementary angles help prove lines are parallel by applying the Consecutive Interior Angles Theorem. This problem demonstrates how angle relationships between lines intersected by a transversal ensure parallelism. Suitable for students in Grades 8-10.

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