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.