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!

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.