1. **Problem statement:** We have triangle KLM with interior angles $\angle M = 50^\circ$, $\angle L = 48^\circ$, and $\angle K = 82^\circ$. Points N, O, and P are midpoints of the sides of triangle KLM, forming triangle NOP inside it. We need to find the measure of $\angle LNO$. 2. **Key property:** When midpoints of the sides of a triangle are connected, the smaller triangle formed (triangle NOP) is similar to the original triangle KLM and each side of NOP is parallel to one side of KLM. Also, the angles in triangle NOP are equal to the corresponding angles in triangle KLM. 3. **Identify angle $\angle LNO$:** Point N is the midpoint of side KL, and point O is the midpoint of side LM. Since N and O are midpoints, segment NO is parallel to side KM of triangle KLM. 4. **Use parallel lines:** Because NO is parallel to KM, $\angle LNO$ corresponds to $\angle LKM$ in triangle KLM. 5. **Find $\angle LKM$:** $\angle LKM$ is the angle at vertex K in triangle KLM, which is given as $82^\circ$. 6. **Conclusion:** Therefore, $\angle LNO = 82^\circ$. **Final answer:** $$\boxed{82^\circ}$$