https://file.aiquickdraw.com/mathbot/file/1729261624764v03djwvp.jpg

From the image, we have an angle diagram and a problem that asks to solve for \( x \). Here's the breakdown:

### Diagram Analysis:
- There are two intersecting lines at point \( G \), creating vertical angles.
- One angle is labeled \( 51^\circ \).
- The other angle is labeled as \( (13x - 1)^\circ \).
  
### Key Geometry Principle:
- **Vertical angles are congruent**, meaning they are equal. Therefore:
\[
51^\circ = (13x - 1)^\circ
\]
We can now solve for \( x \).

### Step-by-step solution:
1. Set up the equation:
\[
13x - 1 = 51
\]
2. Add 1 to both sides:
\[
13x = 52
\]
3. Divide both sides by 13:
\[
x = 4
\]

So, \( x = 4 \).

### Equation to choose:
The correct equation to use is \( 13x - 1 = 51 \), which corresponds to one of the options provided.

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

### 5 Related Questions:
1. What are vertical angles, and how are they identified?
2. Can you explain other angle relationships in intersecting lines?
3. How do we set up equations for complementary angles?
4. What is the sum of angles in a triangle, and how can we use it to solve for unknowns?
5. How do we apply the same principle to parallel lines cut by a transversal?

### Tip:
When working with angle diagrams, always check for vertical angles and linear pairs, as they provide key relationships for setting up equations.

Which equation can be used to solve for x in the given angle diagram, and what is the value of x?

Learn how to solve for the variable x using the Vertical Angles Theorem in an angle diagram. The problem involves solving a linear equation and applying key geometric concepts. Suitable for Grades 7-9.

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