In the image provided, the problem asks for the value of $x$, given that lines $m \parallel n$ and a transversal $t$ intersect both lines. The figure shows an angle of $60^\circ$ at the intersection of $t$ and line $n$, and an angle labeled $x^\circ$ at the intersection of $t$ and line $m$.

Since the lines $m \parallel n$, the angles formed by the transversal and the parallel lines are either corresponding angles or alternate interior angles, both of which are congruent.

In this case, the $x^\circ$ angle is a corresponding angle to the $60^\circ$ angle because they are on the same side of the transversal and occupy corresponding positions relative to the parallel lines. Therefore:

$x = 60^\circ$

Thus, the correct value of $x$ is $60^\circ$.

Would you like any further details or explanations? Here are some related questions to think about:

1. What are corresponding angles, and why are they equal in this case?
2. What are alternate interior angles, and how are they used with parallel lines?
3. What happens to the angles if the lines are not parallel?
4. How do transversals affect the relationship between angles in parallel lines?
5. Can you solve for $x$ if the given angle were different, say $70^\circ$?

Tip: Always remember that corresponding angles are equal when a transversal intersects two parallel lines.