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The diagram shows two parallel lines \(m\) and \(n\), with a transversal line intersecting them. One angle on the transversal is marked as \(164^\circ\), and we are asked to find the value of \(x^\circ\), which is the corresponding angle on the other side of the transversal.

Since \(m \parallel n\), the two marked angles form a pair of *alternate interior angles*, which means they are equal. This is due to the Alternate Interior Angles Theorem.

Therefore, we have:
\[
x = 164^\circ
\]

### Final Answer:
\[
x = 164^\circ
\]

Would you like more details on the reasoning or clarification on the theorems used?

Here are some related questions for further exploration:
1. What are corresponding angles and how are they related in parallel lines?
2. Can you explain the Consecutive Interior Angles Theorem?
3. How do we prove that two lines are parallel using angles?
4. What is the difference between alternate interior and alternate exterior angles?
5. How can transversal lines be used to solve for unknown angles in geometry?

**Tip**: In parallel line geometry, always look for relationships like alternate interior angles, corresponding angles, and consecutive interior angles to solve for unknown values.

Learn how to find the value of x using the Alternate Interior Angles Theorem. This problem involves two parallel lines m and n, with a transversal intersecting them, and an angle given as 164°. Suitable for students in Grades 8-10.

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