The sum of the interior angle measures for any n-sided polyg

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Anaira Mitch: Master | Next Rank: 500 Posts; Posts: 235; Joined: Wed Oct 26, 2016 9:21 pmFollowed by: 5 members; Thanked: 3 times

by Anaira Mitch » Tue Mar 07, 2017 12:40 pm

The sum of the interior angle measures for any n-sided polygon equals 180(n - 2). If Polygon A has interior angle measures that correspond to a set of consecutive integers, and if the median angle measure for Polygon A is 140Â°, what is the smallest angle measure in the polygon?

(A) 130Â°
(B) 135Â°
(C) 136Â°
(D) 138Â°
(E) 140Â°

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Source: Beat The GMAT — Problem Solving | Tue Mar 07, 2017 12:40 pm

DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 amFollowed by: 128 members; Location: Boston, MA; Thanked: 1153 times; GMAT Score:770

by DavidG@VeritasPrep » Tue Mar 07, 2017 3:19 pm

Anaira Mitch wrote:The sum of the interior angle measures for any n-sided polygon equals 180(n - 2). If Polygon A has interior angle measures that correspond to a set of consecutive integers, and if the median angle measure for Polygon A is 140Â°, what is the smallest angle measure in the polygon?

(A) 130Â°
(B) 135Â°
(C) 136Â°
(D) 138Â°
(E) 140Â°

Useful tidbit: in an evenly spaced set, the mean = the median. So we know that if the median is 140, the mean is also 140. If there are n angles, the sum of those angles will be 140n.

Now we know 140n = 180 * (n-2)
140n = 180n - 360
-40n = 360
n = 9

If there are 9 angles that are consecutive integers, and the median is 140, our set will look like this: 136, 137, 138, 139, 140, 141, 142, 143, 144. The smallest angle is 136. The answer is C

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by [email protected] » Tue Mar 07, 2017 9:40 pm

Hi Anaira Mitch,

This question can be solved with 'brute force', although you do have to know a few math/geometry rules to get to the correct answer.

To start, when it comes to polygons, another way to think about the total of the angles in the polygon is: "add a side, add 180 degrees"...

For example:
3-sided shape = 180 degrees
4-sided shape = 360 degrees
5-sided shape = 540 degrees
6-sided shape = 720 degrees
7-sided shape = 900 degrees
Etc.

Here, we're told that a polygon with an unknown number of sides has angles that are CONSECUTIVE INTEGERS and that the MEDIAN of those angles = 140 degrees.

Since we're dealing with consecutive integers, the median angle of the polygon can only be one of two options: an integer (if there are an ODD number of sides) or a non-integer (a number that ends in .5.... if there are an EVEN number of sides). Since the median here is 140, there MUST be an ODD number of sides.

From here, we can brute-force the possibilities until we find the one that fits all of the given information. Keep in mind that we need the sum of the angles to be the proper MULTIPLE of 180...

3-sided shape: 139, 140, 141 - This is clearly not a possible triangle.
5-sided shape: 138, 139, 140, 141, 142 - This totals 700 degrees, which doesn't "match" (a 5-sided shape has 540 degrees)
7-sided shape: 137, 138, 139, 140, 141, 142, 143 - This totals 980 degrees, which doesn't "match" (a 7-sided shape has 900 degrees)
9-sided shape: 136, 137, 138, 139, 140, 141, 142, 143, 144 - This totals 1260 degrees, which DOES "match" (a 7-sided shape has 1260 degrees)

Thus, we're dealing with a 9-sided polygon and the smallest angle is 136 degrees.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Tue Mar 07, 2017 10:13 pm

Anaira Mitch wrote:The sum of the interior angle measures for any n-sided polygon equals 180(n - 2). If Polygon A has interior angle measures that correspond to a set of consecutive integers, and if the median angle measure for Polygon A is 140Â°, what is the smallest angle measure in the polygon?

(A) 130Â°
(B) 135Â°
(C) 136Â°
(D) 138Â°
(E) 140Â°

Hi Anaira Mitch,

We have the sum of the interior angles for the n-sided polygon = 180(n - 2).

Since the median interior angle for the polygon = 140, and the interior angles form a set of consecutive integers, the median would be equal to mean.

Thus, Mean of interior angles = 180(n - 2)/n = 140

=> 180(n - 2) = 140n

=> n = 9

So, this is a 9-sided polygon, whose 5th interior angle = 140; there would be 4 angles that are less than 140 and there would be 4 angles that are greater than 140.

Since the angles have a difference of '1' (they are consecutive integers), the smallest angle would be equal to 140 - 4 = 136.

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Geometry Guide

-Jay
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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Thu Mar 23, 2017 3:21 pm

Anaira Mitch wrote:The sum of the interior angle measures for any n-sided polygon equals 180(n - 2). If Polygon A has interior angle measures that correspond to a set of consecutive integers, and if the median angle measure for Polygon A is 140Â°, what is the smallest angle measure in the polygon?

(A) 130Â°
(B) 135Â°
(C) 136Â°
(D) 138Â°
(E) 140Â°

We need to first determine the number of sides (or angles) of polygon A. Let n denote the number of sides of polygon n. Since the interior angle measures correspond to consecutive integers, the median angle measure is also the average angle measure. Since sum = average x quantity, we have sum = 140n. Since we are also given that the sum of the interior angle measures equals 180(n - 2), it must be true that 140n = 180(n - 2). Thus:

140n = 180(n - 2)

140n = 180n - 360

-40n = -360

n = 9

We now know polygon A is a 9-sided polygon and there must be 4 angles that have measures less than the median angle measure (and 4 angles that have measures greater than the median angle measure). Since the median angle measure is 140 degrees and the angle measures are consecutive integers, the smallest angle measure must be 140 - 4 = 136 degrees.

Answer: C

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri Mar 24, 2017 5:05 am

Here's a related practice question (related to n-sided polygons) for you: https://www.beatthegmat.com/geometry-dec ... 69524.html

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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