Concave polygon Convex polygon Not a polygon is the measure of the third largest angle? 1 degree larger than the angle just smaller than it. What $123.57^{\circ}$ $115.57^{\circ}$ $128.07^{\circ}$ $129.57^{\circ}$

Step 1 ：Let us call $x$ the number of sides in the polygon. The sum of all the angles of the polygon with $x$ sides is $180(x-2)$, but with the information given, it can also be expressed as $123.57 + 115.57(x-1)$.

Step 2 ：Therefore, setting these two equations equal: \begin{align*} 180(x-2) &= 123.57 + 115.57(x-1)\\ 180x - 360 &= 123.57 + 115.57x - 115.57\\ 64.43x &= 237.43\\ x &= 3.68\end{align*}

Step 3 ：However, the number of sides in a polygon must be a whole number, so we round $3.68$ up to $4$.

Step 4 ：Thus, it has \(\boxed{4}\) sides.