The diagram shows two parallel lines, $l$ and $m$, cut by a transversal $p$. The angles formed by the transversal and the parallel lines are labeled as $85^\circ$ and $(5x - 20)^\circ$.

Solution:

Since $l \parallel m$, the corresponding angles formed by the transversal $p$ are congruent. This means that the angle $85^\circ$ is congruent to the angle $(5x - 20)^\circ$.

Thus, we can set up the equation: $85 = 5x - 20$

Now, solve for $x$:

1. Add 20 to both sides: $85 + 20 = 5x$ $105 = 5x$

2. Divide both sides by 5: $x = \frac{105}{5}$ $x = 21$

Thus, the value of $x$ is $\boxed{21}$.

Would you like further clarification or details on this?

Here are 5 related questions to expand your understanding:

1. What are alternate interior angles and how do they relate to parallel lines?
2. How do you prove that corresponding angles are congruent when a transversal cuts parallel lines?
3. What are the other types of angles formed by a transversal and parallel lines (e.g., same-side interior, alternate exterior)?
4. How would the equation change if the angle given was on the same side of the transversal?
5. Can you think of any real-world applications where knowing about angles and transversals is useful?

Tip: Always remember, when a transversal cuts through parallel lines, corresponding angles are congruent, and alternate interior angles are also congruent.