The image shows two parallel lines $m \parallel n$, and a transversal intersecting them. Two angles are formed at the intersection points:

• One angle at the top is labeled as $(x - 30)^\circ$,
• The other angle at the bottom is labeled as $(3x - 10)^\circ$.

Since the lines are parallel and the angles are on opposite sides of the transversal, these are alternate interior angles. For parallel lines, alternate interior angles are equal. Therefore, we can set up the equation:

$x - 30 = 3x - 10$

Solving for $x$:

1. Bring all terms involving $x$ to one side:

$x - 3x = -10 + 30$

2. Simplify the equation:

$-2x = 20$

3. Solve for $x$:

$x = \frac{20}{-2} = -10$

Thus, the value of $x$ is $\boxed{-10}$.

Would you like more details on this solution or further clarifications?

Here are 5 related questions:

1. What are corresponding angles, and how do they relate to parallel lines?
2. How would the solution change if the angles were not alternate interior angles?
3. What is the significance of the transversal in angle relationships with parallel lines?
4. Can you solve for $x$ if the angles were supplementary instead of equal?
5. How do vertical angles relate to the geometry of parallel lines?

Tip: When working with angles formed by parallel lines and a transversal, always check if the angles are alternate, corresponding, or supplementary to guide your solution method.