1. **Problem Statement:** We have an isosceles triangle KLM with sides KL = KM. Points L, M, and S are collinear with M between L and S. The exterior angle at vertex M, adjacent to the triangle, measures 115°. We need to find the measures of angles $\angle L$ and $\angle K$. 2. **Key Concept:** The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two opposite interior angles. Here, the exterior angle at M is $115^\circ$, so: $$\angle L + \angle K = 115^\circ$$ 3. **Isosceles Triangle Property:** Since $KL = KM$, the base angles opposite these sides are equal. Therefore: $$\angle L = \angle K$$ 4. **Set up the equation:** Let $x = \angle L = \angle K$. Then: $$x + x = 115^\circ$$ $$2x = 115^\circ$$ 5. **Solve for $x$:** $$x = \frac{115^\circ}{2} = 57.5^\circ$$ 6. **Final answer:** $$\angle L = 57.5^\circ$$ $$\angle K = 57.5^\circ$$ Thus, each named angle $\angle L$ and $\angle K$ measures 57.5 degrees.