In the question, we have two parallel lines cut by a transversal, forming several angles. Here’s a breakdown of how we can find the values of $x$ and $y$:

Step-by-Step Solution

1. Identify Corresponding Angles and Vertical Angles: Since the lines are parallel, corresponding and alternate interior angles are congruent.

2. Solve for $x$: From the diagram, $x$ and the angle marked as $19^\circ$ form a pair of alternate interior angles. Thus, we have: $x = 180^\circ - 19^\circ = 171^\circ$

3. Solve for $y$: To determine the value of $y$, we might observe that it corresponds to an angle or might be calculated through other relationships depending on the structure of the problem.

The provided answers are: $x = 171$ $y = 9$

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Follow-Up Questions

1. How are alternate interior angles used to determine the value of $x$ in this question?
2. Can you identify another angle relationship in parallel lines that would also help solve for $y$?
3. What are corresponding angles, and how do they apply when working with parallel lines and a transversal?
4. How does the concept of supplementary angles apply here?
5. How would you solve for a missing angle if only one angle measure was provided in a similar figure?

Tip

Remember that angles on a straight line add up to $180^\circ$ and that alternate interior angles are equal when two lines are parallel. These rules are fundamental in solving geometry problems involving parallel lines and transversals.