https://file.aiquickdraw.com/mathbot/file/1725928771455ccmgms75.jpeg

Let's analyze the problem shown in the image.

### Problem Breakdown:
The image depicts two intersecting lines, creating two pairs of vertically opposite angles. The given expressions for these angles are:
- \( 2x + 8^\circ \)
- \( 3x + 17^\circ \)
- \( 5y + 15^\circ \)

Since vertically opposite angles are equal, we can set up the following equations:
1. \( 2x + 8^\circ = 3x + 17^\circ \)  (because the angles opposite to each other must be equal)
2. \( 3x + 17^\circ = 5y + 15^\circ \)

### Solving the Equations:

**1. Solve for \( x \) in the first equation:**
\[
2x + 8^\circ = 3x + 17^\circ
\]
Subtract \( 2x \) from both sides:
\[
8^\circ = x + 17^\circ
\]
Subtract \( 17^\circ \) from both sides:
\[
x = 8^\circ - 17^\circ
\]
\[
x = -9^\circ
\]

**2. Now, solve for \( y \) using the second equation:**
\[
3x + 17^\circ = 5y + 15^\circ
\]
Substitute \( x = -9 \):
\[
3(-9) + 17^\circ = 5y + 15^\circ
\]
Simplify:
\[
-27^\circ + 17^\circ = 5y + 15^\circ
\]
\[
-10^\circ = 5y + 15^\circ
\]
Subtract \( 15^\circ \) from both sides:
\[
-25^\circ = 5y
\]
Divide by 5:
\[
y = -5^\circ
\]

### Final Answer:
- \( x = -9^\circ \)
- \( y = -5^\circ \)

Would you like further details or have any questions? Here are some related questions to consider:

1. What is the definition of vertically opposite angles?
2. How can you determine the relationship between the angles formed by intersecting lines?
3. If a similar problem included a third variable, how would you approach solving it?
4. What do the negative values of \( x \) and \( y \) imply about the angles?
5. How can you verify your solutions for \( x \) and \( y \)?

**Tip:** Remember that vertically opposite angles are always equal, regardless of the values involved.

Find the value of each variable in the figure shown. (2x + 8)°, (3x + 17)°, (5y + 15)°

Learn how to solve for variables x and y in an intersecting lines problem using the Vertically Opposite Angles Theorem. This problem is ideal for students in Grades 8-10 and covers key concepts in algebra and geometry.

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