https://file.aiquickdraw.com/mathbot/file/1727492252056l87cdjda.jpg

In the image, we see two intersecting lines forming vertical angles, with the angle expressions given as \( (8x + 7)^\circ \) and \( (9x - 4)^\circ \).

### Step-by-step solution:

1. **Vertical Angles Theorem**:
   According to this theorem, vertical angles are congruent (equal). Therefore, we can set the expressions for the two angles equal to each other:
   \[
   8x + 7 = 9x - 4
   \]

2. **Solve for \( x \)**:
   - First, subtract \( 8x \) from both sides to isolate \( x \) on one side:
     \[
     7 = 9x - 8x - 4
     \]
     \[
     7 = x - 4
     \]
   - Now, add 4 to both sides to solve for \( x \):
     \[
     7 + 4 = x
     \]
     \[
     x = 11
     \]

Thus, the value of \( x \) is \( 11 \).

Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:
1. What are vertical angles, and why are they always congruent?
2. Can vertical angles ever be supplementary? Why or why not?
3. How would the problem change if the angles were on a straight line?
4. Can we use the same method for solving other angle relationships like corresponding angles?
5. What if the angles given had expressions like \( (2x + 3)^\circ \) and \( (3x + 5)^\circ \)? How would the process differ?

**Tip**: Always double-check your algebraic steps when solving for variables to avoid small mistakes!

Find the value of x in the vertical angles (8x + 7)° and (9x - 4)°.

Learn how to solve for x in the vertical angles equation (8x + 7)° = (9x - 4)°. This geometry problem applies the Vertical Angles Theorem, which states that vertical angles are congruent. A detailed step-by-step solution is provided, suitable for students in Grades 7-10.

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