https://file.aiquickdraw.com/mathbot/file/17260824927574cehuwvu.jpg

In the given image, we are asked to find the value of \( x \), knowing that \( m \parallel n \), meaning lines \( m \) and \( n \) are parallel. 

### Step-by-step Solution:
1. **Recognize the Angle Relationships**: Since \( m \parallel n \), and a transversal crosses them, the angles formed have a known relationship. Specifically, the angle marked \( 130^\circ \) and the angle marked \( x^\circ \) are **consecutive interior angles**.
  
2. **Sum of Consecutive Interior Angles**: For parallel lines cut by a transversal, consecutive interior angles are supplementary. This means their sum is \( 180^\circ \).

   \[
   x + 130^\circ = 180^\circ
   \]

3. **Solve for \( x \)**:

   \[
   x = 180^\circ - 130^\circ
   \]
   \[
   x = 50^\circ
   \]

### Final Answer:
The value of \( x \) is \( 50^\circ \).

Would you like more details on any specific part? Here are 5 related questions to expand this concept:

1. What are alternate interior angles, and how are they used in parallel line geometry?
2. How do we identify corresponding angles when two parallel lines are cut by a transversal?
3. Can you explain why the sum of consecutive interior angles is always \( 180^\circ \) for parallel lines?
4. If the angle opposite to \( 130^\circ \) is given, how would you calculate it?
5. How can you use this information to solve problems involving angles formed by parallel lines and multiple transversals?

**Tip**: In parallel line problems, always check for known angle relationships like alternate interior, corresponding, and consecutive interior angles to make your calculations simpler.

This page provides a solution to a geometry problem where two parallel lines are cut by a transversal, and the goal is to find the value of x using the consecutive interior angles theorem. Suitable for students in Grades 8-10.

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