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The given diagram shows two intersecting lines, forming angles at point \(E\). You are provided with the following angle measures:

- \( m \angle DEH = 9x - 22 \)
- \( m \angle GEF = 6x + 2 \)

Since \(\angle DEH\) and \(\angle GEF\) are **vertical angles** (they are opposite each other), they must be equal. Hence, we can set up the following equation:

\[
9x - 22 = 6x + 2
\]

Now, let's solve this equation for \(x\).

\[
9x - 6x = 2 + 22
\]
\[
3x = 24
\]
\[
x = \frac{24}{3} = 8
\]

Thus, the value of \(x\) is \(8\).

Let me know if you'd like a detailed explanation or have any questions. Here are some related questions you might find interesting:

1. How do you determine if two angles are vertical angles?
2. Can you explain why vertical angles are always equal?
3. What are supplementary angles, and how do they differ from vertical angles?
4. How would you solve this problem if the angles were supplementary?
5. How can we apply this concept to real-world situations involving angles?

**Tip:** When solving angle problems involving intersecting lines, always look for relationships like vertical, supplementary, or complementary angles to set up equations.

The m ∠DEH = 9x - 22 and m ∠GEF = 6x + 2. Find the value of x.

Learn how to solve for x by applying the Vertical Angles Theorem with the equation m∠DEH = 9x - 22 and m∠GEF = 6x + 2. This math problem involves key concepts such as angle relationships and algebra, suitable for students in Grades 7-10.

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