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# The measures of the interior angles in a polygon are consecutive integ

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Joined: 14 Nov 2016
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Re: The measures of the interior angles in a polygon are consecutive integ  [#permalink]

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30 Mar 2017, 18:37
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rxs0005 wrote:
The measures of the interior angles in a polygon are consecutive integers. The smallest angle measures 136 degrees. How many sides does this polygon have?

A) 8
B) 9
C) 10
D) 11
E) 13

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.

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Re: The measures of the interior angles in a polygon are consecutive integ  [#permalink]

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24 Apr 2017, 06:20
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Reverse Approach using external angle

We are given : smallest internal angle = 136 deg ;

to find : number of sides 'n'

Solution : We know that, internal angle = (n - 2)*180;

But, we dont know value of n. However, we know one thing for sure. Irrespective of the value of n, Sum of all the external angles will be 360 deg.

So, Corresponding external angle for internal angle of 136 deg = 180 - 136 = 44 deg. (Since, Sum of internal + external angle = 180 deg)

As internal angle increases by 1 external angle decreases by 1.

So, now 2nd external angle will be 43 deg, 3rd external angle will be 42 deg, 4th will be 41 deg and so on. We keep doing this till the point our sum of all external angles turns out to be 360 deg.

So, 44 + 43 + 42 + 41 + 40 + 39 + 38 + 37 + 36 = 360

total number of terms in above equation is 9. So the number of sides of polygon = 9

( We can also use concept of AP. All the terms are in AP. We know S = 360, t1 = 44, d = -1, n=?
360 = n/2* (2*44 + (n-1)*-1) => n^2 - 89n + 720 = 0 => n = 80 or n = 9)
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Re: The measures of the interior angles in a polygon are consecutive integ  [#permalink]

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02 Sep 2019, 01:53
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).
Re: The measures of the interior angles in a polygon are consecutive integ   [#permalink] 02 Sep 2019, 01:53

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