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The problem shows two intersecting lines with two angles labeled: \( (2x)^\circ \) and \( (4x + 108)^\circ \). Since these two angles are vertical angles (opposite each other at the intersection), they must be equal.

We can set up the following equation based on this information:

\[
2x = 4x + 108
\]

Let's solve for \( x \):

1. Subtract \( 4x \) from both sides:
   \[
   2x - 4x = 108
   \]
   \[
   -2x = 108
   \]

2. Divide by -2:
   \[
   x = \frac{108}{-2} = -54
   \]

So, the value of \( x \) is \( -54 \).

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

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Here are five related questions to expand your understanding:

1. What are vertical angles, and why are they equal?
2. How would the problem change if the angles were complementary instead of vertical?
3. Can you find the measure of both angles using \( x = -54 \)?
4. How do alternate interior angles relate to vertical angles in parallel line problems?
5. What happens if we change the equation to \( (2x + k) = (4x + 108) \)? How would you solve for \( k \)?

**Tip**: Always verify the relationship between angles before setting up equations to avoid common errors.

Find the value of x, given two vertical angles: (2x)° and (4x + 108)°.

Learn how to solve a vertical angles problem, where (2x)° equals (4x + 108)°. This solution breaks down the algebraic process step-by-step for students in Grades 8-10, using the vertical angles theorem.

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