This image shows two intersecting lines, and angles at the intersection are marked with expressions. The angles are labeled as follows:

• One angle is $(3x)^\circ$
• The vertically opposite angle is $(8y - 102)^\circ$
• Another angle is $(2y + 6)^\circ$
• The vertically opposite angle to that is unlabeled but would be supplementary to $(3x)^\circ$, as they are on a straight line.

Using the fact that vertically opposite angles are equal and that supplementary angles add up to $180^\circ$, we can set up the following relationships:

1. $3x = 8y - 102$ (since vertically opposite angles are equal)
2. $3x + (2y + 6) = 180$ (since these angles are supplementary)

Let's solve these equations step by step.

Step 1: Solve the first equation for one variable

$3x = 8y - 102$

Solve for $x$:

$x = \frac{8y - 102}{3}$

Step 2: Substitute this into the second equation

Substitute $x = \frac{8y - 102}{3}$ into $3x + (2y + 6) = 180$:

$3 \left( \frac{8y - 102}{3} \right) + (2y + 6) = 180$

Simplify:

$(8y - 102) + (2y + 6) = 180$

Combine like terms:

$10y - 96 = 180$

Step 3: Solve for $y$

Add 96 to both sides:

$10y = 276$

Now divide by 10:

$y = 27.6$

Step 4: Solve for $x$

Substitute $y = 27.6$ into $x = \frac{8y - 102}{3}$:

$x = \frac{8(27.6) - 102}{3}$

Calculate:

$x = \frac{220.8 - 102}{3} = \frac{118.8}{3} = 39.6$

Final Solution

• $x = 39.6$
• $y = 27.6$

Would you like more details or further clarification on any step? Here are five related questions to explore:

1. How do you determine if two angles are vertically opposite?
2. What is the difference between supplementary and complementary angles?
3. Can you solve for the angle measurements using the values of $x$ and $y$?
4. How would you set up the equations if the problem had given a different angle configuration?
5. What happens if two angles add up to more or less than $180^\circ$?

Tip: Always remember that vertically opposite angles are equal, and adjacent angles on a straight line are supplementary (sum to $180^\circ$). This is a key property in many geometry problems!