1. **Problem Statement:** Find the measure of the minor arc $JN$ given two secants intersecting outside the circle at point $L$ with an angle of $41^\circ$ between them. 2. **Relevant Formula:** The measure of the angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs: $$\text{Angle} = \frac{1}{2} |m(\text{arc } JN) - m(\text{arc } KM)|$$ 3. **Given Data:** - Angle formed at $L$ between secants $LKJ$ and $LMN$ is $41^\circ$. - The angle is split into $41^\circ$ and $47^\circ$ by point $K$. 4. **Step-by-step Solution:** - Let $m(\text{arc } JN) = x$ (the minor arc we want to find). - Let $m(\text{arc } KM) = y$. - Using the formula: $$41 = \frac{1}{2} |x - y|$$ - Multiply both sides by 2: $$\cancel{2} \times 41 = \cancel{2} \times \frac{1}{2} |x - y| \Rightarrow 82 = |x - y|$$ - So, $$|x - y| = 82$$ 5. **Using the angle split:** - The angle at $L$ is split into $41^\circ$ and $47^\circ$, so the total angle is $41^\circ + 47^\circ = 88^\circ$. - Using the same formula for the total angle: $$88 = \frac{1}{2} |m(\text{arc } JN) - m(\text{arc } KM)|$$ - Multiply both sides by 2: $$\cancel{2} \times 88 = \cancel{2} \times \frac{1}{2} |x - y| \Rightarrow 176 = |x - y|$$ 6. **Reconciling the data:** - Since the problem states the angle between the secants is $41^\circ$ and the split is $41^\circ$ and $47^\circ$, the minor arc $JN$ corresponds to the angle $41^\circ$. - Therefore, the measure of minor arc $JN$ is: $$m(\text{arc } JN) = 2 \times 41 = 82^\circ$$ **Final answer:** $$\boxed{82}$$