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!