https://file.aiquickdraw.com/mathbot/file/1726155995242pd81xqre.jpg

The image shows two intersecting lines that form angles labeled \( x^\circ \), \( y^\circ \), and \( 9y^\circ \). These angles are connected by properties of vertical and adjacent angles.

### Step 1: Identify relationships between the angles
- The angles \( x^\circ \) and \( y^\circ \) are vertical angles, so they are equal:
  \[
  x = y
  \]

- The angles \( y^\circ \) and \( 9y^\circ \) are on a straight line, which means they are supplementary (their sum is 180°):
  \[
  y + 9y = 180
  \]

### Step 2: Solve for \( y \)
\[
y + 9y = 180
\]
\[
10y = 180
\]
\[
y = \frac{180}{10} = 18
\]

### Step 3: Solve for \( x \)
Since \( x = y \), we also have:
\[
x = 18
\]

### Final Answer:
- \( x = 18 \)
- \( y = 18 \)

Would you like further explanation on any step or more details?

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Here are some related questions for deeper understanding:
1. What are supplementary angles, and how do they apply here?
2. Why are vertical angles equal?
3. Can you explain the difference between supplementary and complementary angles?
4. What is the significance of intersecting lines in geometry?
5. How would this problem change if one of the angles were missing?

**Tip**: Always remember, vertical angles formed by intersecting lines are congruent!

Find the values of the variables based on the given angles in the intersecting lines diagram.

Solve for the variables in a geometry problem involving intersecting lines. The problem uses vertical angles and supplementary angles to find x and y. Suitable for students in Grades 6-8.

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