https://r2.erweima.ai/mathbot/image/17258203872458x0k9det.jpg

The image shows a geometric diagram with two parallel lines \(L\) and \(L_1\), and a transversal crossing both of them. Several angles are marked, including a 120° angle, a 30° angle, and an unknown angle \(x^\circ\), which you are likely asked to find.

Here’s how to solve for \(x^\circ\):

### Step-by-Step Solution:

1. **Identify angle relationships:**
   - The 120° angle is an exterior angle formed by the transversal and the line \(L\).
   - The angle labeled \(30^\circ\) is one of the interior angles on the same side of the transversal as the 120° angle.
   
2. **Use the fact that interior angles on the same side of a transversal are supplementary:**
   - The angles on the same side of the transversal (the 120° angle and the adjacent angle to it on the other side) must sum to 180°.
   - Therefore, the angle adjacent to the 120° angle is:
   \[
   180^\circ - 120^\circ = 60^\circ
   \]

3. **Find the angle \(x^\circ\):**
   - The angle adjacent to the 60° angle on the lower line (labeled as \(x^\circ\)) is an alternate interior angle to the 30° angle, due to the parallel lines \(L\) and \(L_1\).
   - Since alternate interior angles are equal, we have:
   \[
   x^\circ = 30^\circ
   \]

### Final Answer:
\[
x^\circ = 30^\circ
\]

Would you like any further details or clarifications?

Here are 5 related questions to expand your understanding:
1. What are alternate interior angles, and why are they equal?
2. How would the solution change if the lines were not parallel?
3. Can we find other angles in the diagram based on the given information?
4. How do supplementary angles work when dealing with a transversal?
5. What is the difference between corresponding and alternate interior angles?

**Tip:** When working with parallel lines and transversals, always look for patterns like alternate interior angles, corresponding angles, and supplementary angles!

Find the value of the unknown angle x° given two parallel lines L and L1, and angles 120° and 30° as shown in the diagram.

Learn how to find the unknown angle x° in a geometric diagram involving parallel lines and a transversal. This problem explores key concepts like alternate interior angles and supplementary angles, suitable for students in Grades 6-8.

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