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We are given that two parallel lines are cut by a transversal, and we know that \( m\angle 8 = 97^\circ \). Our task is to find \( m\angle 2 \) and \( m\angle 3 \).

### Step-by-Step Solution:

1. **Relationship between \( \angle 8 \) and \( \angle 2 \):**
   - Angles \( \angle 2 \) and \( \angle 8 \) are **corresponding angles** because they are on the same side of the transversal and the same side of the parallel lines.
   - For parallel lines, corresponding angles are **congruent**, so:
   \[
   m\angle 2 = m\angle 8 = 97^\circ
   \]

2. **Relationship between \( \angle 8 \) and \( \angle 3 \):**
   - Angles \( \angle 8 \) and \( \angle 3 \) are **consecutive interior angles**, or sometimes called **same-side interior angles**.
   - Consecutive interior angles are **supplementary**, meaning their measures add up to \( 180^\circ \). So:
   \[
   m\angle 8 + m\angle 3 = 180^\circ
   \]
   Since \( m\angle 8 = 97^\circ \), we have:
   \[
   97^\circ + m\angle 3 = 180^\circ
   \]
   Solving for \( m\angle 3 \):
   \[
   m\angle 3 = 180^\circ - 97^\circ = 83^\circ
   \]

### Final Answers:
- \( m\angle 2 = 97^\circ \)
- \( m\angle 3 = 83^\circ \)

Would you like further details or clarifications?

Here are 5 related questions you can explore:
1. What are alternate interior angles, and how do they relate to parallel lines?
2. If \( m\angle 1 = 97^\circ \), what is \( m\angle 4 \)?
3. How can you prove that two lines are parallel using angle relationships?
4. What are vertical angles, and how do they relate to this diagram?
5. How do transversal angles behave when the lines are not parallel?

**Tip:** Always remember, corresponding angles are congruent when two parallel lines are cut by a transversal!

Two parallel lines are cut by a transversal. Suppose m∠8 = 97°. Find m∠2 and m∠3.

Learn how to find m∠2 and m∠3 when two parallel lines are cut by a transversal, and m∠8 = 97°. This geometry problem covers angle relationships, including corresponding and supplementary angles. Suitable for students in Grades 7-9.

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