https://file.aiquickdraw.com/mathbot/file/1730339617486783cg9xy.jpg

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.

Learn how to find the values of x and y in a geometric problem involving parallel lines and a transversal. This solution explains corresponding and alternate interior angles and demonstrates how to solve for x and y when m || n. Suitable for students in Grades 8-10.

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