To solve the problem, we are given that $m \parallel n$, meaning lines $m$ and $n$ are parallel. The angles provided are:

• $(9x + 2)^\circ$ and $(5x - 18)^\circ$ are corresponding angles.
• $(2y + 20)^\circ$ is a supplementary angle to $(5x - 18)^\circ$.

Step-by-Step Solution:

1. Since $m \parallel n$, corresponding angles are equal: $9x + 2 = 5x - 18$ Solving for $x$: $9x - 5x = -18 - 2$ $4x = -20$ $x = -5$

2. Now, to find $y$, we use the supplementary relationship: $(5x - 18) + (2y + 20) = 180^\circ$

   Substituting $x = -5$: $(5(-5) - 18) + (2y + 20) = 180$ $(-25 - 18) + (2y + 20) = 180$ $-43 + 2y + 20 = 180$ $2y - 23 = 180$ $2y = 203$ $y = 101.5$

Answer:

• $x = -5$
• $y = 101.5$

Would you like further details or have any questions? Here are five related questions for further exploration:

1. What are corresponding angles, and why are they equal when lines are parallel?
2. How do supplementary angles relate to parallel lines?
3. What other angle pairs are congruent when two parallel lines are intersected by a transversal?
4. How would the calculation change if the lines were not parallel?
5. Can you identify other relationships between angles in parallel line situations?

Tip: Remember, when two parallel lines are cut by a transversal, the angles formed have specific relationships (corresponding, alternate interior, etc.), which can simplify solving for unknowns.