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The diagram shows two parallel lines, \( l \) and \( m \), cut by a transversal, forming two angles labeled \( (8x - 77)^\circ \) and \( (3x + 38)^\circ \). Here's how we can address the questions:

### 1. What angles are given?
The angles \( (8x - 77)^\circ \) and \( (3x + 38)^\circ \) are on opposite sides of the transversal and inside the parallel lines \( l \) and \( m \). This makes them **alternate interior angles**.

### 2. Solve for \( x \).
Since the angles are alternate interior angles, they are equal when the lines are parallel. So, we set the expressions equal to each other:

\[
8x - 77 = 3x + 38
\]

Now, solve for \( x \):

1. Subtract \( 3x \) from both sides:

   \[
   8x - 3x - 77 = 38
   \]

   \[
   5x - 77 = 38
   \]

2. Add 77 to both sides:

   \[
   5x = 115
   \]

3. Divide by 5:

   \[
   x = 23
   \]

### Solution
- The given angles are **alternate interior** angles.
- The value of \( x \) is **23**.

Would you like further details or have any questions?

### Related Questions
1. What are alternate exterior angles, and how do they differ from alternate interior angles?
2. How do you identify corresponding angles in a transversal diagram?
3. What other angle relationships exist when two parallel lines are cut by a transversal?
4. Can consecutive interior angles be supplementary? Explain why.
5. How would the solution change if the lines were not parallel?

### Tip
When working with parallel lines and a transversal, remember that alternate interior angles are always equal, which is a helpful property for solving equations.

Use the diagram to answer the following questions: What angles are given? Solve for x.

Learn how to solve for x in a problem involving parallel lines and transversals. Use the Alternate Interior Angle Theorem to find the value of x and understand angle relationships. Suitable for students in Grades 8-10.

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