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In this problem, we need to prove that angle \( \angle 2 \cong \angle 3 \).

### Given Information:
1. Lines \( a \parallel b \).
2. Lines \( x \parallel y \).

### Explanation of Proof
Since we have two sets of parallel lines (\( a \parallel b \) and \( x \parallel y \)), we can use the properties of parallel lines and transversal angles to establish congruency between \( \angle 2 \) and \( \angle 3 \).

#### Steps:
1. **Identify the Transversal and Corresponding Angles:**
   - Line \( x \) acts as a transversal across the parallel lines \( a \) and \( b \).
   - By the Corresponding Angles Postulate, corresponding angles are congruent when a transversal intersects two parallel lines.
   
2. **Apply the Corresponding Angles Postulate:**
   - Since \( a \parallel b \) and \( x \) is a transversal, \( \angle 2 \) (on line \( a \)) and \( \angle 3 \) (on line \( b \)) are corresponding angles.
   - Therefore, \( \angle 2 \cong \angle 3 \) by the Corresponding Angles Postulate.

### Conclusion
This establishes that \( \angle 2 \cong \angle 3 \) based on the properties of corresponding angles with respect to parallel lines.

Would you like further details, or do you have any specific questions?

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Here are some related questions to explore:
1. What are alternate interior angles, and how can they be used to prove angle congruence?
2. What is the difference between corresponding angles and alternate exterior angles?
3. Can the same proof apply if lines \( a \) and \( b \) were not parallel?
4. How do we prove that lines are parallel given information about angles?
5. What other angle pairs are formed when a transversal cuts two parallel lines?

**Tip:** When two parallel lines are intersected by a transversal, several types of angle pairs (corresponding, alternate interior, alternate exterior) can be used to determine angle congruencies.

Given: a || b and x || y. Prove: ∠2 ≅ ∠3.

Learn how to prove that two angles are congruent using the Corresponding Angles Postulate when two pairs of lines are parallel. This geometry problem is ideal for students in Grades 8-10 and covers concepts related to parallel lines and angle relationships.

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