In the diagram, we are tasked with finding the value of $x$ that makes the lines $m$ and $n$ parallel. We can achieve this by leveraging the properties of parallel lines and their corresponding angles.

Step-by-Step Explanation:

1. Identify the angles:

   • The given angle on line $m$ is $(3x + 5)^\circ$.
   • The given angle on line $n$ is $134^\circ$.

2. Angle relationship for parallel lines: When two lines are parallel, alternate interior angles are equal. Since the angle on line $m$ and the $134^\circ$ angle on line $n$ are alternate interior angles, we can set them equal to each other.

   Thus, we have the equation: $3x + 5 = 134$

3. Solve for $x$: $3x = 134 - 5$ $3x = 129$ $x = \frac{129}{3} = 43$

Final Answer:

The value of $x$ that makes the lines $m \parallel n$ is $\boxed{43}$.

Would you like a more detailed breakdown, or do you have any further questions?

Here are 5 related questions to deepen your understanding:

1. What are alternate interior angles, and why are they equal for parallel lines?
2. How would the problem change if the lines were not parallel?
3. Can you apply this method to other types of angles, like corresponding or vertical angles?
4. What other types of angle relationships occur when two lines are parallel and cut by a transversal?
5. How does solving for $x$ change if the given angles are supplementary?

Tip: When dealing with geometry problems involving parallel lines, always look for relationships like alternate interior, corresponding, or consecutive interior angles to simplify your solution.