https://file.aiquickdraw.com/mathbot/file/1727203913133wq7gh3q8.jpg

In the diagram you provided, we have two parallel lines, \( l \) and \( m \), cut by a transversal, forming angles 58° and \( (5x - 2)° \).

### Step 1: Identifying the relationship of the angles
The two angles shown are on opposite sides of the transversal and are at corresponding positions with respect to the parallel lines \( l \) and \( m \). Thus, these are **corresponding angles**, and by the corresponding angles theorem, corresponding angles are equal when the lines are parallel.

So, we have the equation:
\[
58° = 5x - 2°
\]

### Step 2: Solving for \( x \)
Now, solve the equation for \( x \):

1. Add 2 to both sides:
   \[
   58 + 2 = 5x
   \]
   \[
   60 = 5x
   \]

2. Divide both sides by 5:
   \[
   x = \frac{60}{5} = 12
   \]

Thus, \( x = 12 \).

### Answers:
- The angles are **corresponding**.
- The value of \( x \) is \( 12 \).

Let me know if you need more details, or have any questions!

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Here are five related questions to expand on this:

1. What are alternate interior angles, and how do they differ from corresponding angles?
2. If the angle on the other side of the transversal was \( (7x + 10)° \), how would you solve for \( x \)?
3. What are the properties of consecutive interior angles when dealing with parallel lines?
4. How would you determine the angles if the lines \( l \) and \( m \) were not parallel?
5. What is the relationship between supplementary and corresponding angles in such a diagram?

**Tip:** When working with parallel lines and a transversal, always remember the different angle relationships (corresponding, alternate interior, alternate exterior, and consecutive interior) to set up the correct equations.

Use the diagram to answer the questions in this section.

Explore the concept of corresponding angles with a detailed example involving parallel lines and a transversal. This educational content is aimed at students in Grades 8-10, providing a step-by-step guide on how to identify and solve for angles in a geometric diagram.

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