In the diagram, we observe vertical angles, which are equal to each other. The angle pairings are as follows:

• The angle $7x - 2$° is vertically opposite the angle $18y$°.
• The angle $11x - 34$° is vertically opposite the same angle $18y$°.

Since vertically opposite angles are equal, we can set up the following two equations based on the diagram:

1. $7x - 2 = 18y$
2. $11x - 34 = 18y$

Now, let's solve this system of equations to find the values of $x$ and $y$.

Solving the system:

1. From the first equation: $7x - 2 = 18y \quad \Rightarrow \quad 18y = 7x - 2 \quad \Rightarrow \quad y = \frac{7x - 2}{18}$

2. Substitute this expression for $y$ into the second equation: $11x - 34 = 18y$ Substituting $y = \frac{7x - 2}{18}$ into this: $11x - 34 = 18\left(\frac{7x - 2}{18}\right)$ Simplifying: $11x - 34 = 7x - 2$ Now, solve for $x$: $11x - 7x = -2 + 34$ $4x = 32$ $x = 8$

Finding $y$:

Now that we have $x = 8$, substitute this into the equation for $y$: $y = \frac{7(8) - 2}{18} = \frac{56 - 2}{18} = \frac{54}{18} = 3$

Final Answer:

• $x = 8$
• $y = 3$

Let me know if you'd like further clarification or have any additional questions!

Here are 5 related questions:

1. What are the properties of vertical angles?
2. How can you check the consistency of a system of linear equations?
3. What happens when two lines intersect? Can you always find angle relationships?
4. How do we know when two angles are supplementary or complementary?
5. Can vertical angles be used to solve problems involving parallel lines?

Tip: When dealing with angles, it's helpful to remember that vertical angles are always equal, while adjacent angles on a straight line sum up to 180°.