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The measures of the interior angles in a polygon are

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The measures of the interior angles in a polygon are [#permalink] New post 03 Feb 2011, 14:45
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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
[Reveal] Spoiler: OA

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Re: Geometry [#permalink] New post 03 Feb 2011, 15:02
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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


Sum of Interior Angles of a polygon is 180(n-2) where n is the number of sides (so is the number of angles);

We are told that the smallest angle is 136 degrees, the next will be 136+1 degrees, ..., and the largest one, n_{th} angle, will be 136+(n-1) degrees. The sum of the n consecutive integers (the sum of n angles) is given by \frac{first+last}{2}*# \ of \ terms=\frac{136+(136+n-1)}{2}*n=\frac{271+n}{2}*n;

So we have that 180(n-2)=\frac{271+n}{2}*n --> 360(n-2)=(271+n)*n, now try the answer choices: in order RHS to end with zero (as LHS is because of 360) then n, out of the options listed, could be either 10 or 9, n=9 fits.

Answer: B.
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Re: Geometry [#permalink] New post 03 Feb 2011, 20:05
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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


When I look at this question, I say, "I know how to find the Interior angle of a regular polygon. But this is not a regular polygon since it has angles 136, 137, 138, 139, 140, 141, 142 ....etc."

Also,
Interior angle of 8 sided regular polygon = 180*6/8 = 135
Interior angle of 9 sided regular polygon = 180*7/9 = 140
Interior angle of 10 sided regular polygon = 180*8/10 = 144
etc

The average of the given angles can only match 140 (such that effectively, all the angles are 140) Hence, the polygon must have 9 sides.

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Re: Geometry [#permalink] New post 04 Feb 2011, 09:47
Karishma,
Can you explain in more details when you say "The average of the given angles can only match 140 (such that effectively, all the angles are 140)". I did not get your point. Thanks
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Re: Geometry [#permalink] New post 04 Feb 2011, 10:20
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Sum of interior angles for n sides: (n-2)*180
And each interior angle are increasing by 1.
This can be written as : 136+137+138+139+... = 136+(136+1)+(136+2)+(136+3)+...
For n sides

136n+(1+2+3+4....+n-1)
136n+(n-1)n/2 --- Sum of n-1 natural number

Equate:
136n+(n-1)n/2 = (n-2)*180
n^2-89n+720=0
n^2-80n-9n+720=0
n(n-80)-9(n-80)=0
(n-9)(n-80)=0

n=9
n=80

Since 9 is one of the options.

Ans: "B"
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Re: Geometry [#permalink] New post 04 Feb 2011, 11:56
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ajit257 wrote:
Karishma,
Can you explain in more details when you say "The average of the given angles can only match 140 (such that effectively, all the angles are 140)". I did not get your point. Thanks


Sum of interior angles of a polygon = (n-2)*180 (not necessarily regular polygon)

Why? See the figure below:

Attachment:
Ques2.jpg
Ques2.jpg [ 5.14 KiB | Viewed 4686 times ]

A 6 sided polygon can be split into 4 triangles each of which has a sum of interior angles 180 degrees.
An n sided polygon can be split into n - 2 triangles.

When the polygon is regular, each angle is same so the sum is divided by the number of sides to get the measure of each angle e.g. in a regular hexagon, each interior angle = 4*180/6 = 120 degrees.

Now if I have a hexagon whose angles are 115, 117, 119, 121, 123 and x, what will be the angle x?
We can see it in two ways -
1. The sum of all angles should be 4*180 = 720
So 115 + 117 + 119 + 121 + 123 + x = 720
or x = 125


2. The average of the angles should be 120. (Since the sum of the angles is 720 and there are 6 sides)
119 and 121 average out as 120. (119 is 1 less than 120 and 121 is 1 more than 120)
117 and 123 average out as 120.
So 115 and x should average out as 120 too. Therefore, x should be 125.

In the question, the average of the given angles of the polygon can only be 140. So it must have 9 sides. To confirm,
136, 137, 138, 139, 140, 141, 142, 143, 144 - 9 angles with average 140. So the polygon must have 9 sides.

(It cannot be 144 or anything else because 10 angles (136, 137, 138, 139, 140, 141, 142, 143, 144, 145) will not give an average of 144)
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Re: Geometry [#permalink] New post 05 Feb 2011, 08:01
tough one. simple solution, but its not an ez one.
am i right?
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Re: Geometry [#permalink] New post 06 Feb 2011, 06:25
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144144 wrote:
tough one. simple solution, but its not an ez one.
am i right?


Actually, I wouldn't say it is very tough if we know that the sum of interior angles of a polygon is 180(n -2) (I explained above why it is so). We can also use a very straight forward but long approach.

Interior angles of the given polygon: 136, 137, 138, 139, 140, 141, 142, 143....

Using options:

If the polygon had 8 sides, it would have had 8 interior angles. Their sum: (8-2)180 = 1080
Sum of 8 angles: 136 +137 + 138 + 139 + 140 + 141 + 142 + 143 is more than 1080 hence this polygon does not have 8 sides.

If the polygon had 9 sides, it would have had 9 interior angles. Their sum: (9-2)180 = 1260
Sum of 9 angles: 136 +137 + 138 + 139 + 140 + 141 + 142 + 143 + 144 is 1260. Hence this polygon does have 9 sides.
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Re: Geometry [#permalink] New post 06 Feb 2011, 06:45
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144144 wrote:
tough one. simple solution, but its not an ez one.
am i right?


I wouldn't say it's very hard either, though I wouldn't recommend trial and error in this cases:

...
Calculating the sum of the interior angles of a polygon with 7 sides then checking whether 7 consecutive integers starting from 136 add up to that value;
Calculating the sum of the interior angles of a polygon with 8 sides then checking whether 8 consecutive integers starting from 136 add up to that value;
Calculating the sum of the interior angles of a polygon with 9 sides then checking whether 9 consecutive integers starting from 136 add up to that value;

Whereas equating two formulas and working on answer choices should give an answer in less time: 180(n-2)=\frac{271+n}{2}*n --> 360(n-2)=(271+n)*n --> n=9.
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Re: Geometry [#permalink] New post 06 Feb 2011, 13:03
ye... well i guess i need to work on my geometry more so the way i process these things will be better...

thanks both as usual.
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Re: Geometry [#permalink] New post 09 Aug 2012, 09:08
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Hey guys...please let me know whether this method is correct or not.
the stem says that the angles are consecutive integers, 136 being the smallest one. Also going by the formula of sum of interior angles, we know atleast this much that sum of angles cant be number like 729, 847, 653, 542 but it can be a number of the form xx0. So going by this knowledge, if the smallest angle is 136 then the sum of angles can only be of the form xx0 when the largest angle is 144. That is 9 sides.
Yeah the largest angle could be 154, 164 but in that case the number of sides must be 19 and 29 respectively.
Since the largest option is 13, hence the answer is 9.
Please correct me if I am wrong.
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Re: The measures of the interior angles in a polygon are [#permalink] New post 11 Feb 2014, 08:52
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Re: The measures of the interior angles in a polygon are [#permalink] New post 05 Apr 2014, 15:32
I dunno if this method is valid, please advice

But a polygon has sum of sides equal to (n-2)*180, thus sum will always have 0 as units digit.

Therefore if we begin with 136+137 etc... only 9 fits the bill since 4 is not a valid answer choice

Thanks!
Cheers
J 8-)
Re: The measures of the interior angles in a polygon are   [#permalink] 05 Apr 2014, 15:32
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