https://file.aiquickdraw.com/mathbot/file/17267824147393ap8581a.jpg

The problem in the image shows two intersecting lines, forming vertical angles. The expressions given for the vertical angles are:

- \( (5x + 109)^\circ \)
- \( (11x + 145)^\circ \)

Since vertical angles are congruent (equal), we can set the two expressions equal to each other:

\[
5x + 109 = 11x + 145
\]

### Solving for \( x \):
1. Subtract \( 5x \) from both sides:
   \[
   109 = 6x + 145
   \]

2. Subtract 145 from both sides:
   \[
   109 - 145 = 6x
   \]
   \[
   -36 = 6x
   \]

3. Divide by 6:
   \[
   x = \frac{-36}{6} = -6
   \]

Thus, the value of \( x \) is \( \boxed{-6} \).

Would you like more details on the solution? Here are some questions to further explore this concept:

1. What are vertical angles, and why are they always equal?
2. How can you verify that your solution for \( x \) is correct?
3. What happens if you substitute \( x = -6 \) back into the original angle expressions?
4. What are the properties of intersecting lines and how do they apply to other angle relationships?
5. Can the value of \( x \) ever be a positive number in this particular problem?

**Tip:** Whenever you encounter intersecting lines, remember that vertical angles (the angles opposite each other) are always equal, which helps in solving for unknowns.

Find the value of x for the vertical angles (5x + 109)° and (11x + 145)°.

Learn how to solve for x in the vertical angles equation (5x + 109)° = (11x + 145)°. This algebra problem is perfect for students in Grades 8-10 and covers key concepts like solving linear equations and the vertical angles theorem.

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