1. **State the problem:** We need to find the measure of angle $\angle K$ in triangle $KLM$.\n\n2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always $180^\circ$. That is, \n$$\angle K + \angle L + \angle M = 180^\circ.$$\n\n3. **Analyze the given information:** The problem provides angles for triangle $XYZ$ as $\angle Y = 110^\circ$, $\angle X = 45^\circ$, and $\angle Z = 25^\circ$. These sum to $110 + 45 + 25 = 180^\circ$, confirming the triangle angle sum rule. However, no angles are given for triangle $KLM$.\n\n4. **Conclusion:** Since no angles or other information about triangle $KLM$ are provided, $\angle K$ cannot be determined from the given data.\n\n**Final answer:** The measure of $\angle K$ cannot be found with the information provided.