1. **State the problem:** We need to find the measure of the minor arc $KL$ in a circle where $\angle JNK = 107^\circ$ and $\angle MJN = 99^\circ$. 2. **Understand the setup:** Points $J, K, L, M$ lie on the circle, and $N$ is the center. $\angle JNK$ is a central angle measuring $107^\circ$, so arc $JK$ measures $107^\circ$. Similarly, $\angle MJN = 99^\circ$ is an inscribed angle subtending arc $ML$. The measure of an inscribed angle is half the measure of its intercepted arc. 3. **Find arc $ML$:** Since $\angle MJN = 99^\circ$ is an inscribed angle, the arc $ML$ it intercepts measures $$2 \times 99^\circ = 198^\circ.$$ 4. **Calculate the total circle and remaining arcs:** The total circle measures $360^\circ$. We know arcs $JK = 107^\circ$ and $ML = 198^\circ$. The remaining arcs are $KL$ and $JM$. Since $J, K, L, M$ are points on the circle in order, the arcs $JK + KL + LM + MJ = 360^\circ$. Note that $LM$ and $ML$ refer to the same arc, so $LM = ML = 198^\circ$. 5. **Find arc $KL$:** The arcs $JK + KL + ML = 360^\circ$ (assuming $JM$ is not an arc but a chord). So, $$KL = 360^\circ - JK - ML = 360^\circ - 107^\circ - 198^\circ = 55^\circ.$$ 6. **Conclusion:** The measure of the minor arc $KL$ is $55^\circ$.