In this problem, we need to find the values of angles $x$ and $y$, where two parallel lines are intersected by a transversal.

Step-by-step explanation:

1. Identify the relationships between angles:

   • The angle marked $35^\circ$ and angle $x^\circ$ are corresponding angles because they are on the same side of the transversal and on corresponding positions relative to the parallel lines. Corresponding angles are equal when the lines are parallel.

   • The angle marked $y^\circ$ and $35^\circ$ are alternate interior angles. Alternate interior angles are also equal when two lines are parallel.

2. Set up the equations:

   • Since $x^\circ$ and $35^\circ$ are corresponding angles:
     $x = 35^\circ$

   • Since $y^\circ$ and $35^\circ$ are alternate interior angles:
     $y = 35^\circ$

Final Answer:

$x = 35^\circ \quad \text{and} \quad y = 35^\circ$

Let me know if you would like more details or if you have any further questions.

Here are 5 related questions for you to practice:

1. If two angles are alternate exterior angles, are they congruent when the lines are parallel?
2. What is the sum of the interior angles on the same side of a transversal when two lines are parallel?
3. What is the relationship between corresponding angles when two lines are not parallel?
4. Can corresponding angles be supplementary in any scenario?
5. How do alternate interior angles differ from same-side interior angles?

Tip: For parallel lines, always remember that corresponding angles and alternate interior angles are congruent.