In this problem, you're asked to find the value of $x$ that makes the two lines $m \parallel n$. The two angles provided are $3x + 5^\circ$ and $134^\circ$.

Since $m \parallel n$, the angles form a linear pair because they are on a straight line formed by the transversal intersecting the two parallel lines. Linear pairs are supplementary, meaning their sum is $180^\circ$.

Step-by-step solution:

1. Set up the equation for supplementary angles: $(3x + 5)^\circ + 134^\circ = 180^\circ$

2. Solve for $x$: $3x + 5 + 134 = 180$ Simplify: $3x + 139 = 180$ Subtract 139 from both sides: $3x = 41$ Divide by 3: $x = \frac{41}{3} \approx 13.67$

Final answer:

The value of $x$ that makes $m \parallel n$ is approximately $13.67$.

Would you like further details or have any questions?

Here are five related questions to explore:

1. What are alternate interior angles, and how do they relate to parallel lines?
2. How can you verify if two lines are parallel based on angle relationships?
3. What is the difference between supplementary and complementary angles?
4. How do corresponding angles behave when two lines are parallel?
5. Can you solve for $x$ if the angles were given as alternate interior angles?

Tip: For parallel lines, angle relationships such as corresponding angles, alternate interior angles, and same-side interior angles are key to solving problems.