Let’s solve the problem given in the image.

Problem:

We are tasked to find the value of $x^\circ$, given the following:

• Lines $l$, $m$, and $n$ are all parallel ($l \parallel m \parallel n$).
• The angle adjacent to $x^\circ$ is $65^\circ$.

Solution:

Key Observations:

1. Since $l \parallel m \parallel n$, and the transversal intersects them, the corresponding and alternate interior angles are equal.

2. The angle $x^\circ$ is on the opposite side of the transversal from $65^\circ$. Because the transversal creates alternate interior angles, $x^\circ = 65^\circ$.

Final Answer:

$x = 65^\circ$

Would you like a detailed breakdown of the geometry concepts involved? Here are 5 related questions to expand your understanding:

1. What are alternate interior angles, and why are they equal for parallel lines?
2. How do corresponding angles behave when two lines are parallel?
3. If $l \parallel m \parallel n$, what other relationships can be deduced between the angles in this setup?
4. What happens when the transversal crosses at different angles with parallel lines?
5. Can you find other angles in this diagram if given additional values?

Tip: Always look for alternate interior, corresponding, and supplementary angles when solving parallel line problems with a transversal!