Because this is a convex Pentagon, each angle is less than 180, and because the degree ratio of the inner angle is 1: 2: 3: 4: 8, the maximum angle is 540 * 8/( 1+2+3+4+8) = 240, so there is

Replace 8 with 4, then each angle is 270/7, 540/7, 9 10/7, 1080/7, 1080/7 respectively.