https://file.aiquickdraw.com/mathbot/file/1727463784642au3zx7d3.jpg

The image shows a set of angles formed by two intersecting lines, and we have four expressions representing angles:

1. \( (12x + 8)^\circ \)
2. \( (3x + 17)^\circ \)
3. \( (5y + 15)^\circ \)
4. \( \text{Another angle vertically opposite to } (5y + 15)^\circ \).

Since the angles formed by intersecting lines are either vertically opposite or supplementary, we can use the following relationships:

- Vertically opposite angles are equal.
- Supplementary angles add up to \( 180^\circ \).

### Step 1: Use the supplementary relationship
Let's consider the angles on the straight line, which should add up to \( 180^\circ \). This gives us two equations:

\[
(12x + 8) + (3x + 17) = 180
\]
Simplifying:
\[
15x + 25 = 180
\]
\[
15x = 155
\]
\[
x = \frac{155}{15} = 10.33
\]

### Step 2: Vertically opposite angles
Now, the angles \( (5y + 15)^\circ \) and \( (3x + 17)^\circ \) are vertically opposite, so they must be equal:

\[
5y + 15 = 3x + 17
\]

Substitute the value of \( x = 10.33 \) into the equation:

\[
5y + 15 = 3(10.33) + 17
\]
\[
5y + 15 = 30.99 + 17
\]
\[
5y + 15 = 47.99
\]
\[
5y = 47.99 - 15
\]
\[
5y = 32.99
\]
\[
y = \frac{32.99}{5} = 6.60
\]

### Final values:
- \( x \approx 10.33 \)
- \( y \approx 6.60 \)

Let me know if you'd like further details, clarifications, or calculations!

---
Here are some follow-up questions to deepen your understanding:

1. What is the relationship between vertically opposite angles and supplementary angles?
2. How can you verify if your solution to \( x \) and \( y \) satisfies both equations?
3. Can you explore other angle pairs in this diagram to check their consistency?
4. Why do supplementary angles always sum up to \( 180^\circ \) on a straight line?
5. How can you apply these angle relationships to other geometric figures like triangles or quadrilaterals?

**Tip:** Always double-check if two angles are vertically opposite or supplementary by looking at their position in the diagram.

The image shows a set of angles formed by two intersecting lines, where the angles are represented by expressions: (12x + 8)°, (3x + 17)°, (5y + 15)° and its vertically opposite angle.

This problem involves solving for unknown variables x and y using algebraic expressions to represent angles formed by intersecting lines. The solution demonstrates how to apply the concepts of supplementary and vertically opposite angles to find the values of x and y.

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