1. **Problem Statement:** We have two triangles with vertices J, N, K and K, L, M respectively. Given sides are $JN=15$, $NK=5$, $KL=9$, and $LM=10$. We need to find the missing side(s). 2. **Understanding the problem:** The triangles share vertex K. We are given three sides: two sides of the first triangle ($JN=15$, $NK=5$) and two sides of the second triangle ($KL=9$, $LM=10$). The missing side is likely $JM$ or $KM$ depending on the problem context. 3. **Assuming the triangles are similar or connected:** If the problem implies similarity or a relation, we can use the triangle inequality or proportionality. However, no explicit similarity is given. 4. **Check if the triangles share side $K$ and if $NK$ and $KL$ are connected:** Since $NK=5$ and $KL=9$, and $K$ is common, the side $NL$ might be missing or $JM$. 5. **Without additional information, the missing side in triangle J, N, K is $JK$ and in triangle K, L, M is $KM$.** 6. **If the triangles are connected at K and the problem asks for missing side $JK$ or $KM$, we can use the triangle inequality:** For triangle J, N, K: $$JK < JN + NK = 15 + 5 = 20$$ $$JK > |JN - NK| = |15 - 5| = 10$$ So, $$10 < JK < 20$$ For triangle K, L, M: $$KM < KL + LM = 9 + 10 = 19$$ $$KM > |KL - LM| = |9 - 10| = 1$$ So, $$1 < KM < 19$$ 7. **Final answer:** The missing sides satisfy the inequalities: $$10 < JK < 20$$ $$1 < KM < 19$$ Without more data, exact values cannot be determined.