In this diagram, we have two parallel lines $m \parallel n$ and a transversal that creates corresponding and alternate interior angles. Let's analyze the problem step-by-step.

Step 1: Identify Angle Relationships

Since $m \parallel n$:

1. The angle $(9x - 12)^\circ$ (above line $m$) and $(5x - 4)^\circ$ (below line $n$) are corresponding angles. Therefore, they are equal: $9x - 12 = 5x - 4$

2. $y^\circ$ (above line $m$) and $(5x - 4)^\circ$ (below line $n$) are alternate interior angles. Therefore, they are also equal: $y = 5x - 4$

Step 2: Solve for $x$

Starting with the equation from the corresponding angles: $9x - 12 = 5x - 4$ Rearrange to solve for $x$: $9x - 5x = -4 + 12$ $4x = 8$ $x = 2$

Step 3: Substitute $x$ to Find $y$

Now that we know $x = 2$, substitute it into $y = 5x - 4$: $y = 5(2) - 4$ $y = 10 - 4$ $y = 6$

Solution

The values of $x$ and $y$ are: $x = 2, \quad y = 6$

Would you like a further breakdown of the solution? Here are some related questions for additional practice:

1. What are corresponding angles and how do they relate when lines are parallel?
2. How would the solution change if $m$ and $n$ were not parallel?
3. Can you explain why alternate interior angles are equal when two lines are parallel?
4. What would the value of $y$ be if $x = 3$?
5. How can you determine if two angles are supplementary or complementary in a transversal setup?

Tip: Remember that in a diagram with parallel lines, corresponding angles are always equal, as are alternate interior angles. This rule is crucial for solving many geometric problems involving parallel lines and transversals.