Solution: Let the external angle be x degrees and the number of sides be n, then there is

(n-2)× 180 +x= 1350

180 ×n= 17 10 -x

n =( 17 10-x)/ 180

=9+(90 -x)/ 180

=9-(x-90 )/ 180

Since n is a positive integer, (x-90) must be a multiple of 180 to be divisible, and 0

(1) x-90 = 0, then x = 90, then n=9-0=9, and the number of sides of the polygon is 9;

(2) x-90 = 180, then x = 270, at this time n=9- 1=8, and the number of sides of the polygon is 8;

So the number of sides of this polygon is 8 and 9.