In the given diagram, we are tasked with finding the value of $x$ such that lines $m \parallel n$ (lines $m$ and $n$ are parallel). The diagram shows two angles formed by a transversal cutting through these two lines: one angle is $(3x - 28)^\circ$, and the other is $(2x + 35)^\circ$.

Since the lines are parallel, the corresponding angles formed by the transversal must be equal. Thus, we can set up the equation:

$3x - 28 = 2x + 35$

Now, let's solve for $x$:

1. Subtract $2x$ from both sides: $3x - 2x - 28 = 35$ $x - 28 = 35$

2. Add 28 to both sides: $x = 35 + 28$ $x = 63$

So, the value of $x$ is $63$.

Would you like more details, or do you have any questions about this solution?

Here are some related questions to further explore this concept:

1. What are alternate interior angles, and how do they relate to parallel lines?
2. If the lines were not parallel, how would that affect the relationship between the angles?
3. How can we prove that corresponding angles are congruent when lines are parallel?
4. What other types of angles are formed when a transversal cuts parallel lines?
5. How would the solution change if the angle measures were given in radians instead of degrees?

Tip: In problems involving parallel lines, always look for relationships like corresponding, alternate interior, and alternate exterior angles to set up equations for solving.