In this problem, we are given two parallel lines $m \parallel n$, and a transversal that intersects them, forming angles. Specifically, one of the angles is $164^\circ$, and we need to find the value of $x$ (the angle adjacent to $164^\circ$).

Step-by-Step Solution:

1. Identify the relationship between the angles: Since $m \parallel n$, we can apply the rule of supplementary angles on a straight line. Adjacent angles formed by a transversal and a line add up to $180^\circ$.

2. Set up the equation: The angle $x^\circ$ and $164^\circ$ are adjacent angles, so: $x^\circ + 164^\circ = 180^\circ$

3. Solve for $x$: $x = 180^\circ - 164^\circ = 16^\circ$

Thus, the value of $x$ is $16^\circ$.

Would you like further details on any step or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How do you calculate corresponding angles when a transversal crosses parallel lines?
2. What is the difference between alternate interior and alternate exterior angles?
3. Can you explain the concept of vertically opposite angles?
4. How do you identify and use co-interior angles in a similar problem?
5. What are some common errors to avoid when solving problems with parallel lines and transversals?

Tip: Always remember that adjacent angles on a straight line sum up to $180^\circ$, which is useful when dealing with transversals cutting through parallel lines.