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In the given diagram, lines \( \overleftrightarrow{MO} \) and \( \overleftrightarrow{PR} \) are parallel, and you are given that the measure of \( \angle MNL = 55^\circ \). The task is to find the measure of \( \angle MNQ \).

### Step-by-step solution:
1. **Identify the angle relationship:**
   - Since \( \overleftrightarrow{MO} \parallel \overleftrightarrow{PR} \), and \( \overleftrightarrow{ML} \) acts as a transversal, the angles \( \angle MNL \) and \( \angle MNQ \) form a pair of **alternate interior angles**. 

2. **Apply the alternate interior angle theorem:**
   - By the alternate interior angle theorem, if two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Hence, 
   \[
   \angle MNL = \angle MNQ
   \]

3. **Conclusion:**
   - Since \( \angle MNL = 55^\circ \), we conclude that:
   \[
   \angle MNQ = 55^\circ
   \]

Thus, \( m \angle MNQ = 55^\circ \).

Would you like more details on any part of this explanation?

### Related questions:
1. What is the alternate interior angle theorem?
2. How can we identify alternate interior angles in a diagram?
3. Are corresponding angles also congruent when a transversal cuts two parallel lines?
4. Can you explain the properties of transversal lines?
5. How would the solution change if the lines were not parallel?

### Tip:
When working with parallel lines and transversals, always look for alternate interior, corresponding, and vertical angles—they often provide key relationships for solving geometry problems.

If MO and PR are parallel lines and m∠MNL = 55°, what is m∠MNQ?

Learn how to find the measure of ∠MNQ when lines MO and PR are parallel, and m∠MNL is given as 55°. This geometry problem involves parallel lines and transversals, using the alternate interior angle theorem for the solution. Suitable for students in Grades 8-10.

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