We are tasked with solving for $x$, given that lines $m$ and $n$ are parallel.

Key Observations:

1. Since $m \parallel n$, the angles formed by the transversal with these lines have relationships based on parallel line angle properties, such as corresponding angles and alternate interior angles.
2. The angle marked $(x+2)^\circ$ is alternate interior to the angle $111^\circ$, meaning they are equal: $x + 2 = 111$

Solving the Equation:

$x + 2 = 111$ $x = 111 - 2$ $x = 109$

Final Answer:

$x = 109$

Would you like me to explain any part of the reasoning in more detail?

Related Questions:

1. What are the key angle relationships formed by a transversal crossing two parallel lines?
2. How do alternate interior angles differ from corresponding angles?
3. Can the value of $x$ change if the lines $m$ and $n$ are not parallel? Why or why not?
4. What other angles in the diagram can be calculated once $x$ is known?
5. How can we prove that lines $m$ and $n$ are parallel using geometry?

Tip:

Always look for clues like parallel lines and transversal angles to quickly determine relationships between angles!