Okay, let's see. I need to find the number of sides in a convex polygon where the sum of all but one of its interior angles is 1070 degrees. Hmm, convex polygon, so all interior angles are less than 180 degrees, right? And the sum of the interior angles of a polygon with n sides is given by the formula (n-2)*180 degrees. But here, the problem says the sum of all but one of the interior angles is 1070 degrees. So that means if the total sum is S, then S minus one angle equals 1070. Therefore, S = 1070 + x, where x is the measure of that one angle that's excluded. 

But since it's a convex polygon, each interior angle must be less than 180 degrees. So x < 180. Therefore, S must be less than 1070 + 180, which is 1250 degrees. Also, S must be greater than 1070 degrees because x is positive. So 1070 < S < 1250. 

But S is also equal to (n-2)*180. So we can write (n-2)*180 must be between 1070 and 1250. Let me set up the inequality:

1070 < (n-2)*180 < 1250

Divide all parts by 180 to solve for n:

1070 / 180 < n - 2 < 1250 / 180

Calculating the left side: 1070 ÷ 180. Let me do that. 180*5=900, 1070-900=170. So 5 + 170/180 = 5 + 17/18 ≈ 5.944. So the left side is approximately 5.944.

Similarly, 1250 ÷ 180. 180*6=1080, 1250-1080=170. So 6 + 170/180 = 6 + 17/18 ≈ 6.944.

Therefore, n - 2 is between approximately 5.944 and 6.944. Since n must be an integer (number of sides can't be a fraction), n - 2 must be 6. Therefore, n = 8. Wait, wait, 5.944 < n-2 < 6.944. So n - 2 could be 6, but also, since 5.944 is approximately 5.944, which is just below 6. So n-2 must be 6. Because 6 is the integer between 5.944 and 6.944. Therefore, n = 8.

But let me check. If n = 8, then the total sum of interior angles is (8-2)*180 = 6*180 = 1080 degrees. Then, the sum of all but one angle is 1070, so the excluded angle is 1080 - 1070 = 10 degrees. Which is less than 180, so that's valid for a convex polygon. So that works. 

But wait, let me verify if there could be another n. Suppose n-2 is 6, so n=8. If n-2 were 7, then n=9. Then total sum would be (9-2)*180=1260. Then the excluded angle would be 1260 - 1070 = 190. But 190 is greater than 180, which would make the polygon non-convex. So that's not allowed. Hence, n must be 8.

Alternatively, if n-2 were 5, then n=7. Then total sum is (7-2)*180=900. Excluded angle would be 900 - 1070 = -70, which is impossible. So n can't be 7. Therefore, the only possible n is 8.

Wait, but let me check again. Let me compute (n-2)*180 for n=8: 6*180=1080. 1080 - 1070=10, which is okay. For n=9: 7*180=1260, 1260-1070=190. Not okay. For n=7: 5*180=900, 900-1070=-170. Not okay. So yes, only n=8 works.

Therefore, the answer is 8.

But wait, let me think again. The problem says "the sum of all but one of its interior angles is 1070 degrees". So the total sum is 1070 + x, where x is the missing angle. Since all interior angles in a convex polygon are less than 180, x < 180. So 1070 < total sum < 1070 + 180 = 1250. The total sum is (n-2)*180. So (n-2)*180 must be in (1070, 1250). Let's compute (n-2)*180 for integer n.

We can write 1070 < (n-2)*180 < 1250

Divide by 180: 1070/180 ≈5.944 < n-2 < 1250/180≈6.944

Thus n-2 must be 6 since it's the only integer between 5.944 and 6.944. Hence n=8.

Therefore, the answer is 8.
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