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The problem provided involves a figure with intersecting lines, where we need to find the values of the variables \(x\) and \(y\). Let me break it down step-by-step.

### Part A: 
We are given the following angles:

- \( (8x - 10)^\circ \)
- \( (6y + 20)^\circ \)
- \( (7x)^\circ \)

#### Step 1: Identify the relationships between the angles.
From the diagram, we can observe the following angle relationships:

- \( (8x - 10)^\circ \) and \( (7x)^\circ \) are alternate interior angles (since the lines are parallel), so they are equal. Therefore:
  \[
  8x - 10 = 7x
  \]
  Solving this equation for \(x\):
  \[
  8x - 7x = 10 \implies x = 10
  \]

#### Step 2: Use the value of \(x\) to find the angles involving \(y\).
Next, \( (6y + 20)^\circ \) and \( (7x)^\circ \) are vertically opposite angles, which are equal. Substituting \(x = 10\) into \( (7x)^\circ \):
\[
7x = 7(10) = 70^\circ
\]
Now, set this equal to \( (6y + 20)^\circ \):
\[
6y + 20 = 70
\]
Solving for \(y\):
\[
6y = 70 - 20 \implies 6y = 50 \implies y = \frac{50}{6} \implies y = 8.33
\]

So, the values are:
\[
x = 10, \quad y \approx 8.33
\]

### Part B: 
In this part, you are asked to explain the reasoning.

- **Alternate interior angles** were used to equate \( (8x - 10)^\circ \) and \( (7x)^\circ \), which gave us the equation \( 8x - 10 = 7x \).
- **Vertically opposite angles** were used to equate \( (6y + 20)^\circ \) and \( 70^\circ \) to find the value of \(y\).

Would you like more details on any part of this solution? 

Here are five related questions to consider:
1. What are alternate interior angles, and why are they equal?
2. How do we determine when two angles are vertically opposite?
3. If the lines weren't parallel, how would the relationships between the angles change?
4. What are some other geometric properties of parallel lines and transversals?
5. How can we use angle relationships to solve more complex geometric problems?

Tip: When solving geometry problems involving parallel lines, always look for angle relationships like alternate interior, corresponding, or vertically opposite angles.

Find the values of the variables in the given figure involving intersecting lines and alternate interior angles.

Learn how to solve for variables x and y using angle relationships such as alternate interior angles and vertically opposite angles. This geometry problem involves parallel lines and basic algebraic manipulation. Suitable for students in Grades 7-9.

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