https://file.aiquickdraw.com/mathbot/file/17292248838250o35rlgi.jpg

In the image, we are given two intersecting lines forming angles, and we need to solve for \(x\). 

Here’s the step-by-step process:

1. The angle \( \angle KOP \) is labeled as \( (4x + 38)^\circ \), and \( \angle NPJ \) is labeled as \( 118^\circ \).
2. \( \angle KOP \) and \( \angle NPJ \) form a pair of **vertically opposite angles**, which means they are equal. This gives us the equation:
   \[
   4x + 38 = 118
   \]
3. Now, solve for \(x\):
   \[
   4x = 118 - 38
   \]
   \[
   4x = 80
   \]
   \[
   x = \frac{80}{4} = 20
   \]

Thus, \(x = 20\).

Would you like further details or explanations on this? 

Here are 5 follow-up questions you might find useful:
1. What are vertically opposite angles and why are they equal?
2. How can you identify corresponding or alternate angles in parallel line diagrams?
3. What are other properties of intersecting lines regarding angle measures?
4. How can you use algebra to solve for variables in angle equations?
5. What other types of angle relationships can be used to solve similar problems?

**Tip:** Always check the geometry of diagrams for special relationships like vertical angles, alternate interior angles, or corresponding angles—they often simplify problems!

Solve for x in the diagram with two intersecting lines where angle KOP is labeled (4x + 38)° and angle NPJ is 118°.

Learn how to solve for x by using the concept of vertically opposite angles. This problem includes algebraic equations and geometry concepts, suitable for students in Grades 7-9.

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