The problem requires us to find the values of $x$ and $y$ based on the given parallel lines $m \parallel n$ and the relationships between the angles.

Step 1: Identify angle relationships

Since $m \parallel n$, the corresponding angles are equal: \text{Angle y^\circ} = \text{Angle 6x - 12^\circ}. Additionally, the two marked angles form a straight line, so: $y + (9x + 12) = 180.$

Step 2: Write the equations

1. From the corresponding angles: $y = 6x - 12.$

2. From the linear pair: $y + 9x + 12 = 180.$

Step 3: Substitute $y = 6x - 12$ into the second equation

Substitute $y = 6x - 12$ into $y + 9x + 12 = 180$: $(6x - 12) + 9x + 12 = 180.$

Step 4: Simplify and solve for $x$

$6x - 12 + 9x + 12 = 180,$ $15x = 180,$ $x = 12.$

Step 5: Solve for $y$

Using $y = 6x - 12$, substitute $x = 12$: $y = 6(12) - 12,$ $y = 72 - 12 = 60.$

Final Answer:

$x = 12, \quad y = 60.$

Would you like further clarification or a breakdown of the concepts?

Related Questions:

1. How do corresponding angles and linear pair angles work for parallel lines?
2. What are the steps to solve equations with substitution?
3. How would this change if $m$ and $n$ were not parallel?
4. What happens if another angle measure is given instead of one equation?
5. Why is it important to confirm the relationship between angles before solving?

Tip: Always label angle relationships clearly in diagrams to avoid confusion when setting up equations.