https://file.aiquickdraw.com/mathbot/file/1725420412077k15q85e6.jpg

The image depicts a pair of intersecting lines forming four angles, with three of the angles given in terms of algebraic expressions:

- \( (2x + 8)^\circ \)
- \( (3x + 17)^\circ \)
- \( (5y + 15)^\circ \)

Since these are angles formed by intersecting lines, we know that:

1. Vertically opposite angles are equal.
2. Adjacent angles are supplementary (they add up to \(180^\circ\)).

### Step-by-step Solution:

1. **Vertically Opposite Angles**:
   - \( (2x + 8)^\circ \) and \( (3x + 17)^\circ \) are vertically opposite, so:
   \[
   2x + 8 = 3x + 17
   \]

2. **Solving for \(x\)**:
   \[
   2x + 8 = 3x + 17
   \]
   Subtract \(2x\) from both sides:
   \[
   8 = x + 17
   \]
   Subtract 17 from both sides:
   \[
   x = -9
   \]

3. **Using Adjacent Angles (Supplementary)**:
   \( (2x + 8)^\circ \) and \( (5y + 15)^\circ \) are adjacent, so:
   \[
   (2x + 8) + (5y + 15) = 180
   \]
   Substitute \(x = -9\) into the equation:
   \[
   2(-9) + 8 + (5y + 15) = 180
   \]
   Simplify:
   \[
   -18 + 8 + 5y + 15 = 180
   \]
   \[
   5y + 5 = 180
   \]
   Subtract 5 from both sides:
   \[
   5y = 175
   \]
   Divide by 5:
   \[
   y = 35
   \]

### Final Values:
- \(x = -9\)
- \(y = 35\)

Would you like more details, or do you have any questions?

Here are 5 related questions to expand on this topic:
1. What is the measure of each angle after substituting the values of \(x\) and \(y\)?
2. How do you determine the relationships between vertically opposite angles?
3. What is the definition of supplementary angles?
4. Can you find the fourth angle in the diagram, and why is it important?
5. How do changes in the values of \(x\) and \(y\) affect the overall diagram?

**Tip**: Always double-check your algebraic steps to avoid errors when solving for variables.

Determine the values of x and y for the intersecting lines where the angles are given by (2x + 8)°, (3x + 17)°, and (5y + 15)°.

This problem involves finding the values of x and y using algebraic expressions representing angles formed by intersecting lines. The solution demonstrates the use of theorems related to vertically opposite and supplementary angles, making it suitable for students in Grades 8-10.

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