https://file.aiquickdraw.com/mathbot/file/1729453395945lqklp6dy.jpg

In this image, we are given two angles formed by the intersection of two straight lines. The two expressions for the angles are:

- \( (8x - 4)^\circ \)
- \( (9x + 2)^\circ \)

These angles are opposite each other, which means they are *vertical angles* and thus must be equal. Therefore, we can set up the equation:

\[
8x - 4 = 9x + 2
\]

### Step-by-Step Solution:

1. **Set up the equation:**
   \[
   8x - 4 = 9x + 2
   \]

2. **Move all terms involving \(x\) to one side:**
   Subtract \(8x\) from both sides:
   \[
   -4 = x + 2
   \]

3. **Isolate \(x\):**
   Subtract 2 from both sides:
   \[
   -4 - 2 = x
   \]
   \[
   x = -6
   \]

So, the value of \(x\) is \( \boxed{-6} \).

Would you like more details on this or have any other questions?

Here are 5 related questions to deepen your understanding:
1. What are vertical angles and why are they always equal?
2. What would the angles measure if \(x = -6\)?
3. How can you check if the two expressions are really equal for \(x = -6\)?
4. How do alternate interior angles compare to vertical angles?
5. What are the properties of supplementary angles, and how do they differ from vertical angles?

**Tip:** When solving for \(x\) in angle problems, always check if the angles are related (e.g., complementary, supplementary, or vertical) to set up the right equation.

Find the value of x for the given vertical angles (8x - 4)° and (9x + 2)°.

Learn how to solve for x in a geometry problem involving vertical angles. The two angles given are (8x - 4)° and (9x + 2)°. Understand the use of the Vertical Angles Theorem and algebraic equations to find the solution. Suitable for Grade 8-10 students.

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