OFFICIAL SOLUTION

We are told that the smallest angle measures 136 degrees--this is the first term in the consecutive set. If the polygon has S sides, then the largest angle--the last term in the consecutive set--will be (S - 1) more than 136 degrees.

The sum of consecutive integers = (Average Term) * (# of Term)= \(\frac{First + Last}{2}\) * (# of Terms).

Given that there are S terms in the set, we can plug in for the first and last term as follows:

\(\frac{136 + 136 + (S-1)}{2} * S\) = sum of the angles in the polygon.

We also know that the sum of the angles in a polygon = 180 (S-2) where S represents the number of sides.

Therefore: \(180(S-2) = \frac{136+136+(S-1)}{2} * S\). We can solve

for S by cross-multiplying and simplifying as follows:

\(2(180)(S-2) = [272 + (S-1)] S\)
\(360S - 720 = (271 + S)S\)
\(360S - 720 = 271S + S^2\)
\(S^2 - 89S + 720 = 0\)

A look at the answer choices tells you to try (S - 8), (S - 9), or (S - 10) in factoring.

As it turns out (S-9)(S-80)=0, which means S can be 9 or 80. However S cannot be 80 because this creates a polygon with angles greater than 180.

Therefore S equals 9; there are 9 sides in the polygon.

The correct answer is B.
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I still think that the fastest (though may not be the best way as expounded extensively in this post) to solve this question is to make use of the fact that 180(n-2) will always have Unit Digit = 0 (try n=3,4,...etc)

The fact is that we are given a starting number of 136, and we know for SURE that all angles are consecutive numbers. That would mean (pun intended) that the mean * n must = UD of 0.

We can narrow down to 9 or 10, but a surer bet would be to use the fact that the MEAN would always have a UD of zero i.e. only 140 is the nearest possible answer we have here. Then, multiplying by number of sides would always yield UD zero.

Hence (B).