To solve for $x$ given that $m \parallel n$, we need to use the properties of parallel lines and the transversal $t$.

Solution:

1. Since $m \parallel n$, the angles formed by the transversal $t$ with these lines are either corresponding, alternate interior, or alternate exterior angles.

2. In this case, the angle labeled $13^\circ$ and the angle labeled $x^\circ$ are alternate interior angles because they are on opposite sides of the transversal $t$ and inside the parallel lines $m$ and $n$.

3. For parallel lines, alternate interior angles are equal. Therefore: $x = 13^\circ$

Answer:

$x = 13^\circ$

Would you like more details on this solution or have any questions?

Here are some related questions to further explore this concept:

1. What are alternate exterior angles, and how are they used in similar problems?
2. How would you calculate $x$ if the angle was given as $2x + 5$ degrees?
3. Can you apply the same logic to find angles when the lines are not parallel?
4. What other angle relationships exist when dealing with parallel lines and a transversal?
5. How do we determine if two lines are parallel given only angle measurements?

Tip: Remember that when dealing with parallel lines and a transversal, corresponding angles and alternate interior/exterior angles are key relationships to identify for solving unknowns.