jmc

Suppose we list the angle measures in increasing order. Let $x$ be the measure of the middle angle in degrees. Then, the five angles have measures $x-2^{\circ }$, $x-1^{\circ }$, $x$, $x+1^{\circ }$, and $x+2^{\circ }$. The sum of the angle measures in a pentagon is $180(5-2)=540$ degrees, so $$(x-2^{\circ })+(x-1^{\circ })+(x)+(x+1^{\circ })+(x+2^{\circ })=540^{\circ }.$$ Simplifying the left side gives $5x=540^{\circ }$, and dividing both sides by 5 gives $x=108^{\circ }$. Therefore, the largest angle has measure $x+2^{\circ }=\boxed{110^{\circ }}$.