The problem presented involves a pair of parallel lines cut by a transversal, and the task is to find the value of $x$.

Based on the diagram:

• We have two parallel lines, denoted by the parallel symbol ($\parallel$).
• A transversal line intersects the parallel lines, creating various angles.
• The given angle measures are:
  • An angle of $130^\circ$ at the top intersection of the transversal.
  • An angle of $4x^\circ$ and $2x^\circ$ at the bottom intersection.

Explanation:

1. Corresponding Angles: The angle $130^\circ$ at the top intersection is corresponding to the angle $4x^\circ$ at the bottom because corresponding angles are congruent when a transversal cuts two parallel lines.

   Therefore: $4x = 130$ Solving for $x$: $x = \frac{130}{4} = 32.5$

Thus, the value of $x$ is $32.5^\circ$.

Would you like more details or have any other questions?

Here are some related questions:

1. What are the properties of corresponding angles?
2. How do alternate interior angles work when a transversal cuts parallel lines?
3. What is the significance of supplementary angles in transversal geometry?
4. How can we verify if lines are parallel using angle relationships?
5. What are the other types of angles formed by a transversal?

Tip: When solving for unknown angles in geometry problems involving parallel lines, always look for corresponding, alternate interior, and supplementary angles.