In the given diagram, the lines $l$, $m$, and $n$ are parallel ($l \parallel m \parallel n$), and two angles are given: $(9x - 19)^\circ$ and $35^\circ$. We need to find the value of $x$.

Step-by-step solution:

1. Since $l \parallel m \parallel n$, the angles formed by the transversal and the parallel lines follow the properties of alternate interior angles or corresponding angles.

2. The angle marked as $(9x - 19)^\circ$ and the angle marked as $35^\circ$ are corresponding angles. Corresponding angles are equal when two parallel lines are cut by a transversal.

3. Therefore, we can set up the following equation: $9x - 19 = 35$

4. Solve for $x$: $9x = 35 + 19$ $9x = 54$ $x = \frac{54}{9} = 6$

Thus, the value of $x$ is 6.

Let me know if you need more details or explanations. Here are five related questions for further practice:

1. What are alternate interior angles, and how are they different from corresponding angles?
2. If the value of $x$ was different, how would the solution change?
3. Can two parallel lines ever create unequal corresponding angles?
4. What would happen if the transversal was perpendicular to the parallel lines?
5. How do you prove that two lines are parallel using corresponding angles?

Tip: In geometry, recognizing angle relationships such as corresponding, alternate interior, and exterior angles is key to solving problems involving parallel lines.