In the image, we are asked to find the value of $x$ using the given diagram. The diagram shows two intersecting lines, and the angles are marked: one as $36^\circ$ and the other as $12x^\circ$. These two angles are vertical angles, meaning they are equal.

Since vertical angles are congruent, we can set up the equation: $36^\circ = 12x^\circ$

Now, solve for $x$: $12x = 36$ $x = \frac{36}{12} = 3$

Thus, the value of $x$ is $\boxed{3}$.

Would you like more details or have any questions?

Here are some related questions you can consider:

1. What are the properties of vertical angles?
2. How do parallel lines affect angle relationships?
3. Can vertical angles be supplementary, and under what conditions?
4. What other angle relationships exist when two lines intersect?
5. How would the problem change if the angles were supplementary instead?

Tip: Always remember that vertical angles are equal, regardless of the orientation of the intersecting lines.