This problem involves parallel lines $m \parallel n$ with a transversal intersecting them, forming angles labeled by algebraic expressions. To solve for $x$ and $y$, we need to apply the properties of parallel lines and transversals, specifically corresponding angles and linear pairs.

1. Corresponding angles are equal when a transversal crosses two parallel lines: $(9x + 2)^\circ = (5x - 18)^\circ$
2. Alternate interior angles between the parallel lines are also equal: $(9x + 2)^\circ = (2y + 20)^\circ$

Step 1: Solve for $x$

Using the corresponding angles property: $(9x + 2) = (5x - 18)$ Simplify the equation: $9x - 5x = -18 - 2$ $4x = -20$ $x = -5$

Step 2: Solve for $y$

Using the relationship between the angles formed by the transversal: $(9x + 2) = (2y + 20)$ Substitute $x = -5$: $9(-5) + 2 = 2y + 20$ $-45 + 2 = 2y + 20$ $-43 = 2y + 20$ $2y = -43 - 20$ $2y = -63$ $y = -31.5$

Final Answer:

• $x = -5$
• $y = -31.5$

Would you like further details or clarifications?

Here are 5 related questions to deepen understanding:

1. What are the properties of alternate interior angles?
2. How do you recognize corresponding angles in a transversal diagram?
3. Can two angles add up to more than 180° in a linear pair? Why or why not?
4. How would the problem change if $m$ and $n$ were not parallel?
5. What happens to the angle values if $x$ or $y$ turn out to be positive?

Tip: When solving for angles with parallel lines and a transversal, always look for pairs of angles that are either corresponding, alternate interior, or form linear pairs.