To solve for $x$, we will use the concept of vertical angles and linear pairs.

Step-by-Step Solution:

1. Identify the given angles:

   • $\angle KOP = 9x + 30^\circ$
   • $\angle NOP = 12x^\circ$

   Since these two angles are vertical angles, they are congruent (equal).
   Therefore: $9x + 30 = 12x$

2. Solve for $x$:
   Start by isolating $x$:
   $9x + 30 = 12x$ Subtract $9x$ from both sides: $30 = 3x$ Divide both sides by $3$: $x = 10$

Final Answer:

$x = 10$

Would you like me to verify this solution further or explain additional properties of vertical angles?

Related Questions:

1. How do vertical angles relate to each other in geometry?
2. What are alternate interior angles, and when do they appear?
3. How can you solve for missing angles in a triangle?
4. What are supplementary angles, and how are they identified?
5. Can you explain corresponding angles and their properties in parallel lines?

Tip: Vertical angles are always congruent, which is a useful property in solving many geometry problems.