Prove that alternate interior angles 3 and 6 are congruent

To prove that alternate interior angles \( \angle 3 \) and \( \angle 6 \) are congruent, we'll use the properties of parallel lines cut by a transversal.

### Given:
- Lines \( m \) and \( n \) are parallel.
- Line \( t \) is a transversal that intersects lines \( m \) and \( n \).
- Angles are labeled as shown in the diagram: \( \angle 1 \), \( \angle 2 \), \( \angle 3 \), \( \angle 4 \), \( \angle 5 \), \( \angle 6 \), \( \angle 7 \), \( \angle 8 \).

### To Prove:
- \( \angle 3 \cong \angle 6 \).

### Proof Steps:

1. **Corresponding Angles Postulate**:
   - Since \( m \parallel n \) and \( t \) is a transversal, corresponding angles are congruent.
   - \( \angle 2 \cong \angle 6 \) (Corresponding Angles)

2. **Vertical Angles Theorem**:
   - Vertical angles are congruent.
   - \( \angle 2 \cong \angle 3 \) (Vertical Angles)

3. **Transitive Property of Congruence**:
   - Since \( \angle 2 \cong \angle 6 \) and \( \angle 2 \cong \angle 3 \), by the Transitive Property of Congruence, \( \angle 3 \cong \angle 6 \).

### Conclusion:
Thus, by the Corresponding Angles Postulate, the Vertical Angles Theorem, and the Transitive Property of Congruence, we have proven that alternate interior angles \( \angle 3 \) and \( \angle 6 \) are congruent.

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Do you want further details or have any questions? 

Here are some related questions for further practice:

1. How would you prove that \( \angle 4 \) and \( \angle 5 \) are congruent?
2. What role does the Corresponding Angles Postulate play in geometry?
3. Can you identify other pairs of alternate interior angles in the diagram?
4. What is the significance of the Transitive Property in geometric proofs?
5. How can you prove that \( \angle 1 \) and \( \angle 8 \) are congruent?
6. What would happen if lines \( m \) and \( n \) were not parallel?
7. Explain the importance of the Vertical Angles Theorem in this proof.
8. Can you apply the same proof technique to a different set of angles in the diagram?

**Tip**: Always verify that the lines are parallel before applying the Corresponding Angles Postulate or the Alternate Interior Angles Theorem.

Learn how to prove that alternate interior angles \( \angle 3 \) and \( \angle 6 \) are congruent using the Corresponding Angles Postulate, Vertical Angles Theorem, and Transitive Property of Congruence. Suitable for students in Grades 8-10 studying geometry and angles.

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