To solve the problem:

Step-by-Step Solution:

1. Identify key relationships in the diagram:

• The lines $m$ and $n$ are parallel ($m \parallel n$).
• The angles $65^\circ$ and $(4x - 17)^\circ$ form a pair of corresponding angles since the transversal intersects $m$ and $n$.
• Corresponding angles are equal when the lines are parallel.

2. Set up the equation:

Since $65^\circ = (4x - 17)^\circ$: $4x - 17 = 65$

3. Solve for $x$:

$4x = 65 + 17$ $4x = 82$ $x = \frac{82}{4}$ $x = 20.5$

Final Answer:

$x = 20.5$

Would you like a detailed explanation about corresponding angles or the rules for parallel lines and transversals?

Here are 5 related questions to deepen understanding:

1. What are alternate interior angles, and how are they related when lines are parallel?
2. If one angle is $(6x + 5)^\circ$ and the corresponding angle is $95^\circ$, what is $x$?
3. How do supplementary angles relate to parallel lines and a transversal?
4. Can the value of $x$ be negative in problems involving angles? Why or why not?
5. What happens to corresponding angles if the lines are not parallel?

Tip: Always verify the given relationships (e.g., parallel lines or angle pairs) to set up the correct equation in geometry problems!