In this image, we are given two parallel lines, $m \parallel n$, intersected by a transversal, and the question asks us to find the value of $x$.

The angles formed by the transversal cutting parallel lines $m$ and $n$ are either alternate interior angles or corresponding angles. These types of angles are congruent.

We have the following angle expressions:

• One angle on the left: $(5x - 10)^\circ$
• One angle on the right: $(6x + 3)^\circ$

Because these angles are alternate interior angles (formed by the transversal), we can set the expressions equal to each other:

$5x - 10 = 6x + 3$

Now, let's solve for $x$.

First, subtract $5x$ from both sides:

$-10 = x + 3$

Next, subtract 3 from both sides:

$-13 = x$

So, the value of $x$ is $\boxed{-13}$.

Would you like further clarification on how to approach this type of problem or any other related questions?

Follow-up questions:

1. What are corresponding angles, and how are they used in this type of problem?
2. How can you tell when to use alternate interior angles versus corresponding angles?
3. What would happen if the lines were not parallel in this case?
4. What are some other types of angle relationships formed by a transversal cutting parallel lines?
5. How do you check the solution to an angle-based equation like this one?

Tip: Always double-check whether the angles you are working with are congruent or supplementary when dealing with parallel lines and a transversal.