To solve for the variables $x$ and $y$ in the given figure, we can use the properties of angles formed by parallel lines and a transversal. Specifically, the figure shows two sets of angles that can be identified as either corresponding angles or vertically opposite angles.

Step 1: Analyze the angles

1. The angle marked as $(7x - 4)^\circ$ and the angle $(6x + 2)^\circ$ are vertically opposite angles, which means they are equal. $7x - 4 = 6x + 2$

2. The angle marked $(2y + 6)^\circ$ and the angle $(7x - 4)^\circ$ are corresponding angles, meaning they are also equal. $2y + 6 = 7x - 4$

Step 2: Solve the equations

Solve for $x$:

From the vertically opposite angles equation: $7x - 4 = 6x + 2$ Subtract $6x$ from both sides: $x - 4 = 2$ Add 4 to both sides: $x = 6$

Solve for $y$:

Now substitute $x = 6$ into the equation for corresponding angles: $2y + 6 = 7x - 4$ Substitute $x = 6$: $2y + 6 = 7(6) - 4$ $2y + 6 = 42 - 4$ $2y + 6 = 38$ Subtract 6 from both sides: $2y = 32$ Divide by 2: $y = 16$

Final Answer:

• $x = 6$
• $y = 16$

Would you like more details on any specific part of the solution?

Here are five related questions to explore this concept further:

1. What are the different types of angles formed by parallel lines and a transversal?
2. How do corresponding angles relate to each other in parallel line setups?
3. What is the difference between alternate interior and corresponding angles?
4. Can vertically opposite angles ever be unequal? Why or why not?
5. What other methods can be used to solve for variables in geometry problems involving angles?

Tip: Always look for angle relationships like corresponding, alternate, or vertically opposite angles when working with parallel lines and transversals!