jmc

There are 9 vertices in the polygon, so from each vertex there are potentially 8 other vertices to which we could extend a diagonal. However, 2 of these 8 points are connected to the original point by an edge, so they are not connected by interior diagonals. So each vertex is connected to 6 other points by interior diagonals. This gives a preliminary count of $9\times 6=54$ interior diagonals. However, we have counted each diagonal twice (once for each of its endpoints), so we must divide by 2 to correct for this overcounting, and the answer is $\frac{9\times 6}{2}=\boxed{27}$ diagonals.