In the image, we are asked to find the value of $x$ for which lines $m \parallel n$ (where $m$ is parallel to $n$).

Step-by-step solution:

1. Identify angle relationships:

   • The angles $(3x - 34)^\circ$ and $(2x + 20)^\circ$ are corresponding angles because $m \parallel n$, and corresponding angles formed by a transversal with parallel lines are equal.

2. Set up the equation: Since the angles are equal, we set their expressions equal to each other: $3x - 34 = 2x + 20$

3. Solve for $x$:

   • Subtract $2x$ from both sides: $x - 34 = 20$
   • Add 34 to both sides: $x = 54$

Answer:

The value of $x$ for which $m \parallel n$ is $\boxed{54}$.

Would you like more details or have any other questions?

Here are some related questions to explore:

1. What is the significance of corresponding angles when two lines are parallel?
2. How do you verify that two lines are parallel using angle measurements?
3. Can this method be used if the angles were alternate interior angles?
4. What if the transversal intersected the lines at different angles—how would the equation change?
5. How do other types of angle pairs (like consecutive interior angles) help in proving lines parallel?

Tip: Always check the type of angles formed when a transversal crosses parallel lines—corresponding, alternate interior, or consecutive interior—to set up correct equations.