In the given image, we have two lines, $m$ and $n$, which are parallel, and a transversal that crosses them forming an angle of $160^\circ$ with line $n$. The angle marked as $x^\circ$ on the image corresponds to the alternate interior angle relative to the $160^\circ$ angle on the other side of the transversal.

By the Alternate Interior Angles Theorem, which states that alternate interior angles are equal when two lines are parallel and cut by a transversal, we can determine that:

$x = 160^\circ$

Hence, the value of $x$ is $160^\circ$.

Would you like more details or do you have any other questions on this topic? Here are some follow-up questions you might consider:

1. How would the solution change if the angle was not given and only the relationship between the lines was provided?
2. What happens to the angles if the lines $m$ and $n$ were not parallel?
3. How does the Corresponding Angles Postulate apply to parallel lines cut by a transversal?
4. Can you find the angle between the transversal and line $m$ using the given information?
5. What are supplementary and complementary angles in the context of parallel lines and a transversal?

Tip: Remember, when dealing with transversals and parallel lines, always consider the various angle relationships like corresponding angles, alternate interior angles, and consecutive interior angles.