https://file.aiquickdraw.com/mathbot/file/17303875643091egufidq.jpg

To solve for \( x \), we need to analyze the angles given in the diagram. The two angles, \( (6x + 22)^\circ \) and \( (64 - x)^\circ \), are vertical angles, meaning they are equal because they are formed by the intersection of two straight lines. Therefore, we can set up the equation:

\[
6x + 22 = 64 - x
\]

Now, let's solve for \( x \):

1. Add \( x \) to both sides to get:
   \[
   7x + 22 = 64
   \]

2. Subtract 22 from both sides:
   \[
   7x = 42
   \]

3. Divide by 7:
   \[
   x = 6
   \]

So, the solution is:
\[
x = 6
\]

Would you like further details on this solution, or do you have any other questions?

### Related Questions:
1. How are vertical angles identified in different geometric setups?
2. What properties do alternate interior angles have?
3. Can vertical angles ever be supplementary?
4. How can we verify solutions in angle-based equations?
5. What are the common mistakes when solving equations involving angles?

### Tip:
When dealing with intersecting lines, always look for vertical or supplementary angles to set up your equations easily.

Solve for x using the angle relationship in the given diagram.

Learn how to solve for x using vertical angle relationships in intersecting lines. This geometry problem involves setting up and solving an algebraic equation. Ideal for students in Grades 7-9.

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