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To solve for \( x \) in this figure where \( AB \parallel FG \), we will use the following steps:

1. **Understand the Geometry**:
   - The line \( AB \) is parallel to \( FG \).
   - We see two triangles, one smaller (triangle \( ABC \)) and another formed by the lines with angles provided.
   - The figure involves transversal angles created by the parallel lines \( AB \parallel FG \), and the transversal \( DE \) cuts these lines.

2. **Corresponding and Supplementary Angles**:
   - Since \( AB \parallel FG \) and transversal \( DE \) crosses both, the corresponding angles are equal.
   - \( \angle ABE = 2y \) and \( \angle DCE = 5x \), meaning that these angles are related because of the parallelism and the transversal.

3. **Use of Supplementary Angles**:
   - The angle \( \angle EFG = 150^\circ \) is supplementary to the interior angle \( \angle ABC \).
   - Since \( AB \parallel FG \), the angle formed by the transversal at point \( B \), \( \angle ABC \), plus the external angle \( \angle EFG = 150^\circ \), should sum up to \( 180^\circ \). Therefore:
   \[
   \angle ABC + 150^\circ = 180^\circ \implies \angle ABC = 30^\circ.
   \]

4. **Solving for \( x \)**:
   - The corresponding angle property tells us \( x = 30^\circ \).
   
Thus, the value of \( x \) is \( 30^\circ \).

Would you like further clarification on any of these steps?

### Related Questions:
1. What is the relationship between parallel lines and corresponding angles?
2. How do supplementary angles help in solving problems involving transversals?
3. Can the transversal theorem be used with other geometric shapes?
4. Why do parallel lines create corresponding angle relationships with transversals?
5. What other angle properties can be used in triangles formed by parallel lines?

### Tip:
Always look for corresponding and alternate angles when dealing with parallel lines and transversals. These properties can simplify your approach to solving for unknown angles.

En la figura AB // FG, ¿Cuál es el valor de x?

Learn how to find the value of x in a geometric figure involving parallel lines AB and FG, using the transversal theorem, supplementary, and corresponding angles. This problem is ideal for students in Grades 8-10 and covers core geometric principles.

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