https://file.aiquickdraw.com/mathbot/file/1726092026030pz9lcdi7.jpg

The given diagram appears to show two intersecting lines forming four angles. To find the values of \(x\) and \(y\), we can use the properties of vertically opposite angles and linear pairs.

1. **Vertically Opposite Angles**:
   - Vertically opposite angles are equal. In the diagram:
     - \( (3x)^\circ \) and \( (8y - 102)^\circ \) are vertically opposite angles.
     \[
     3x = 8y - 102
     \]

2. **Linear Pair**:
   - Angles on a straight line sum up to 180°. In the diagram:
     - \( (2y + 6)^\circ \) and \( (3x)^\circ \) form a linear pair.
     \[
     (2y + 6) + 3x = 180
     \]
     Simplifying:
     \[
     2y + 6 + 3x = 180
     \]
     \[
     2y + 3x = 174
     \]

Now we have a system of two equations:

\[
1) \ 3x = 8y - 102
\]
\[
2) \ 2y + 3x = 174
\]

We can solve this system to find the values of \(x\) and \(y\).

I'll calculate the values for you.The solutions to the system of equations are:

\[
x = \frac{198}{5} = 39.6
\]
\[
y = \frac{138}{5} = 27.6
\]

Thus, the values of \(x\) and \(y\) are:

\[
x = 39.6^\circ
\]
\[
y = 27.6^\circ
\]

Would you like more details on the steps, or do you have any other questions?

Here are some related questions to expand your understanding:
1. What are the properties of vertically opposite angles?
2. How do linear pairs help in solving angle problems?
3. Can this method be applied to non-intersecting angles?
4. What if the lines were parallel—how would the angle relationships change?
5. How can this technique be extended to solve for unknown angles in polygons?

**Tip:** Always look for geometric properties like vertically opposite angles or linear pairs when solving angle problems, as they often simplify complex diagrams.

Find the values of x and y. (The image shows two intersecting lines with the angles labeled as (3x)°, (8y - 102)°, (2y + 6)°.)

Learn how to find the values of x and y using the properties of vertically opposite angles and linear pairs in geometry. This problem covers key geometric concepts, such as opposite angles and adjacent angles summing to 180°, with a detailed step-by-step solution. Suitable for students in Grades 8-10.

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