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In this diagram, we are given that \( m \angle 1 = m \angle 2 \), which indicates that the two angles have equal measures.

From the diagram, we observe that line \( t \) intersects two parallel lines \( p \) and \( q \), creating various angles. Angles 1 and 2 are on opposite sides of the transversal \( t \), and they are called *corresponding angles*.

### Conclusion:
Since \( m \angle 1 = m \angle 2 \), we can conclude that lines \( p \) and \( q \) are **parallel**. This follows from the *corresponding angles postulate*, which states that if a transversal intersects two lines such that corresponding angles are equal, the lines must be parallel.

Would you like further details or have any questions?

Here are 5 related questions to expand on the topic:
1. What other angle relationships exist in the diagram (e.g., alternate interior, alternate exterior)?
2. How would the situation change if \( m \angle 1 \neq m \angle 2 \)?
3. Can you identify any supplementary angles in the diagram?
4. What theorem guarantees that lines are parallel when corresponding angles are equal?
5. How would you prove two lines are parallel using alternate interior angles instead?

**Tip**: In diagrams with parallel lines, always check the relationships between corresponding, alternate interior, and alternate exterior angles!

In the diagram, m∠1 = m∠2. What can you conclude?

This geometry problem demonstrates how to use the Corresponding Angles Postulate to prove that two lines are parallel. It involves identifying and analyzing angles formed by a transversal intersecting two lines. Suitable for students in Grades 8-10.

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