The problem provides two expressions involving angles formed by intersecting lines. Let’s analyze and solve for $x$ and $y$ using the relationships between the angles.

1. Identify Angle Relationships:

   • The given expressions in the figure are for angles formed by intersecting lines, suggesting that certain angles are either vertically opposite or supplementary.
   • Vertically opposite angles are equal, while adjacent angles along a straight line are supplementary (sum up to 180°).

2. Set Up Equations:

   • The angle labeled $(8x - 10)^\circ$ is vertically opposite to the angle labeled $(6y + 20)^\circ$, so: $8x - 10 = 6y + 20$
   • The angle labeled $(7x)^\circ$ is vertically opposite to the angle labeled $(8x - 10)^\circ$, so: $7x = 8x - 10$

3. Solve the Second Equation for $x$: $7x = 8x - 10$ $-x = -10$ $x = 10$

4. Substitute $x = 10$ into the First Equation to Find $y$: $8(10) - 10 = 6y + 20$ $80 - 10 = 6y + 20$ $70 = 6y + 20$ $50 = 6y$ $y = \frac{50}{6} = \frac{25}{3} \approx 8.33$

Solution:

• $x = 10$
• $y = \frac{25}{3} \approx 8.33$

Would you like a detailed breakdown of each step, or do you have further questions?

Here are some related questions to deepen understanding:

1. How do we determine if two angles are vertically opposite?
2. What makes two angles supplementary?
3. Could these angles form a different relationship in another context?
4. What other properties can we use when solving for unknowns in intersecting lines?
5. How would the solution change if different angle measures were given?

Tip: Always identify relationships between angles (vertical, supplementary, etc.) first, as this often simplifies solving the problem.