In this problem, we are given two parallel lines with a transversal intersecting them, and we need to find the value of $x$. Here’s the process to approach it:

1. Identify Corresponding Angles:

   • From the diagram, angle $5x + 35^\circ$ (denoted as angle 2) corresponds to the angle labeled as $45^\circ$ (denoted as angle 5) because they are alternate interior angles.

2. Set up an equation: Since alternate interior angles are equal when lines are parallel: $5x + 35^\circ = 45^\circ$

3. Solve for $x$: Subtract $35^\circ$ from both sides: $5x = 45^\circ - 35^\circ$ $5x = 10^\circ$

   Now, divide both sides by 5: $x = \frac{10^\circ}{5} = 2^\circ$

Thus, the value of $x$ is $\boxed{2^\circ}$.

Would you like more details on alternate interior angles or any other part of this process?

Here are some related questions for further practice:

1. How are alternate interior angles identified when two parallel lines are intersected by a transversal?
2. If $x = 2^\circ$, what is the actual measure of the angle labeled $5x + 35^\circ$?
3. What is the relationship between corresponding angles when parallel lines are involved?
4. What other angle relationships exist between parallel lines and a transversal (e.g., supplementary angles)?
5. How would the solution change if the transversal were not perpendicular to the parallel lines?

Tip: Always check for congruent angles when solving problems involving parallel lines and transversals.