https://file.aiquickdraw.com/mathbot/file/1726841970178f7exrwo4.jpg

We are given two parallel lines, \( m \parallel n \), and asked to find the values of \( x \) and \( y \) using the angles provided in the image. Let’s break down the problem:

### Step 1: Identify the relationships between angles
Since \( m \parallel n \) and the transversal intersects these lines, the following angle relationships apply:
1. **Corresponding angles**: Angles on the same side of the transversal and in corresponding positions are equal.
2. **Alternate interior angles**: Angles on opposite sides of the transversal but inside the parallel lines are equal.

### Step 2: Set up equations
From the image:
1. The angles \( (9x + 2)^\circ \) and \( (2y + 20)^\circ \) are **corresponding angles**, so:
   \[
   9x + 2 = 2y + 20
   \]

2. The angles \( (9x + 2)^\circ \) and \( (5x - 18)^\circ \) are **alternate interior angles**, so:
   \[
   9x + 2 = 5x - 18
   \]

### Step 3: Solve for \( x \)
We first solve the equation involving \( x \):
\[
9x + 2 = 5x - 18
\]
Subtract \( 5x \) from both sides:
\[
4x + 2 = -18
\]
Subtract 2 from both sides:
\[
4x = -20
\]
Divide by 4:
\[
x = -5
\]

### Step 4: Solve for \( y \)
Substitute \( x = -5 \) into the first equation:
\[
9(-5) + 2 = 2y + 20
\]
Simplify:
\[
-45 + 2 = 2y + 20
\]
\[
-43 = 2y + 20
\]
Subtract 20 from both sides:
\[
-63 = 2y
\]
Divide by 2:
\[
y = -31.5
\]

### Final answer:
- \( x = -5 \)
- \( y = -31.5 \)

Would you like further details on any part of this solution?

### Here are 5 questions to deepen your understanding:
1. How do corresponding angles help in solving parallel line problems?
2. What other angle properties are commonly used in problems involving parallel lines?
3. Could we have used another pair of angles to solve for \( x \) and \( y \)?
4. How would the solution change if the lines were not parallel?
5. Why are alternate interior angles always equal when lines are parallel?

### Tip:
Always double-check the relationships between angles in diagrams involving parallel lines, as they frequently provide the key to solving such problems.

Given m || n, find the value of x and y. Angles involved: (9x + 2)°, (5x - 18)°, and (2y + 20)°.

In this problem, learn how to solve for x and y using corresponding and alternate interior angles. The given angles include (9x + 2)°, (5x - 18)°, and (2y + 20)°, with parallel lines cut by a transversal. Suitable for Grades 9-10 students studying parallel line angle relationships and algebraic equations.

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