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The image shows a problem involving angles formed by intersecting lines. The expression for the angles is given as:

- \( 2(5x - 5)^\circ \)
- \( (5y + 5)^\circ \)
- \( (7y - 9)^\circ \)
- \( (6x + 50)^\circ \)

Since the angles are formed by intersecting lines, we can use the property of vertical angles, which states that vertically opposite angles are equal.

### Step-by-step solution:

1. **Match vertical angles:**
   - \( 2(5x - 5)^\circ \) and \( (6x + 50)^\circ \) are vertically opposite, so:
     \[
     2(5x - 5) = 6x + 50
     \]
   - \( (5y + 5)^\circ \) and \( (7y - 9)^\circ \) are vertically opposite, so:
     \[
     5y + 5 = 7y - 9
     \]

2. **Solve for \( x \):**
   \[
   2(5x - 5) = 6x + 50
   \]
   Expand the left side:
   \[
   10x - 10 = 6x + 50
   \]
   Move all terms involving \( x \) to one side:
   \[
   10x - 6x = 50 + 10
   \]
   \[
   4x = 60
   \]
   Divide by 4:
   \[
   x = 15
   \]

3. **Solve for \( y \):**
   \[
   5y + 5 = 7y - 9
   \]
   Move all terms involving \( y \) to one side:
   \[
   5y - 7y = -9 - 5
   \]
   \[
   -2y = -14
   \]
   Divide by -2:
   \[
   y = 7
   \]

### Final answer:
- \( x = 15 \)
- \( y = 7 \)

Would you like further details or have any questions about the solution?

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Here are 5 related questions for further exploration:
1. What other types of angle relationships exist for intersecting lines?
2. How can you verify your solutions in geometry problems involving angles?
3. What is the significance of the "vertical angle theorem" in geometry?
4. How would the solution change if the expressions for the angles were different?
5. Can angles of intersecting lines always be solved algebraically?

**Tip:** Always ensure you clearly label corresponding angles to avoid mixing them up when using properties like vertical angles.

Find the values for x and y given the expressions for vertically opposite angles: 2(5x - 5)°, (6x + 50)°, (5y + 5)°, (7y - 9)°.

Learn how to solve for x and y by using the vertical angle theorem with expressions like 2(5x - 5)° and (6x + 50)°. This step-by-step guide helps students understand the key algebraic and geometric principles involved. Suitable for Grades 8-10.

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