In this diagram, lines $l$ and $m$ are parallel, and the angles $(9x - 4)^\circ$ and $(5x + 16)^\circ$ are formed by a transversal cutting across these lines.

Since the lines are parallel, these angles are corresponding angles. For corresponding angles, we know that they are congruent (equal in measure). Therefore, we can set the two expressions for the angles equal to each other:

$9x - 4 = 5x + 16$

Solving for $x$:

1. Subtract $5x$ from both sides: $9x - 5x - 4 = 16$ $4x - 4 = 16$

2. Add 4 to both sides: $4x = 20$

3. Divide both sides by 4: $x = 5$

Thus, the value of $x$ is $\boxed{5}$.

Would you like further details on this solution or have any questions?

Here are five related questions to expand on this topic:

1. What are the other types of angle relationships formed by parallel lines and a transversal?
2. Can you explain what alternate interior angles are?
3. How do same-side interior angles behave in a transversal setup with parallel lines?
4. If the given angles were supplementary instead, how would the equation change?
5. What would happen if the lines $l$ and $m$ were not parallel?

Tip: When working with parallel lines and transversals, always look for angle relationships such as corresponding, alternate interior, or alternate exterior angles.