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To solve this problem, we will first recognize that both triangles \( \triangle KJN \) and \( \triangle LNM \) share the segment \( NM \) and that this involves using trigonometry and potentially the Law of Cosines or the Law of Sines.

### Step-by-step approach:

#### Triangle \( \triangle KJN \):
- This triangle has sides of lengths 18, 15, and 12. We can solve this using the **Law of Cosines** to find the angle \( \angle K \).
  
Using the Law of Cosines:
\[
\cos K = \frac{a^2 + b^2 - c^2}{2ab}
\]
Where \( a = 15 \), \( b = 12 \), and \( c = 18 \) (the opposite side to angle \( K \)).

#### Triangle \( \triangle LNM \):
- We are given that one side is 20, and we need to find \( x \), which is the other side. Once we find the angles, we can again use trigonometric identities to solve for \( x \).

Let me first calculate the missing angles using the Law of Cosines for both triangles and proceed from there. I'll calculate that now.The angle \( \angle K \) in triangle \( KJN \) is approximately \( 82.82^\circ \).

Now that we have this angle, we can use the Law of Sines to find the angle \( y \) in triangle \( LNM \), and then apply it to find \( x \). I will proceed with this next step.The angles in triangle \( KJN \) are approximately:
- \( \angle N \approx 41.41^\circ \)
- \( \angle J \approx 55.77^\circ \)

Next, let's use these results to work on triangle \( LNM \) and find the side \( x \) using trigonometric relationships. I'll continue with these calculations.Now, to solve for \( x \) in triangle \( LNM \), we need to apply the **Law of Cosines** or the **Law of Sines**, depending on the information available. From triangle \( KJN \), we know some angles, and now we can find the angles or sides needed for \( LNM \) since it's related through shared geometry.

Let me work through that geometric relationship between the triangles and calculate \( x \). I will calculate this now.The next step requires examining if there is enough data from the given triangle \( KJN \) to help directly with the triangle \( LNM \). However, it seems this requires more clarification on angles or potential missing relations between triangles.

Would you like to clarify more details on angle relations between these triangles, or should I focus on explaining further geometric relations to move forward? 

Let me know if you'd like a breakdown of the angles and how they relate!

Determine the value of x using the given side lengths and angles in the two connected triangles KJN and LNM.

Learn how to determine the value of side x in triangle LNM using trigonometry, including the Law of Cosines and the Law of Sines. This problem involves analyzing two connected triangles with given side lengths and angles, making it ideal for high school students studying advanced geometry.

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