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In the image, the lines \( l \), \( m \), and \( n \) are parallel, as indicated by the notation \( l \parallel m \parallel n \). There is a transversal intersecting these lines, forming angles, including the given angle of \( 137^\circ \) and the unknown angle \( x^\circ \).

Using the fact that \( l \parallel m \parallel n \), we can apply the properties of alternate interior angles and supplementary angles formed by a transversal cutting through parallel lines.

### Solution:

1. The angle \( 137^\circ \) and angle \( x^\circ \) are **supplementary angles**, as they are on the same side of the transversal. Supplementary angles add up to \( 180^\circ \).

Thus, we can write the equation:
\[
x + 137^\circ = 180^\circ
\]

2. Solving for \( x \):
\[
x = 180^\circ - 137^\circ = 43^\circ
\]

Thus, the value of \( x \) is \( 43^\circ \).

Would you like more details or have any questions? Here are some related questions you might consider:

1. What are alternate interior angles and how do they relate to parallel lines?
2. How do corresponding angles behave when a transversal intersects parallel lines?
3. What is the significance of supplementary angles in geometry?
4. Can you explain the concept of vertically opposite angles?
5. How do parallel line properties apply in real-world structures or designs?

**Tip:** When working with parallel lines and transversals, it's useful to remember the various angle relationships like alternate interior, corresponding, and supplementary angles.

Solve for the angle x using the property of supplementary angles formed by a transversal intersecting parallel lines l, m, and n. This problem demonstrates how to apply the supplementary angles theorem in geometry, ideal for students in Grades 7-10.

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