Problem Solving

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

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 140°, what is the smallest angle measure in the polygon?

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

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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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 140°, what is the smallest angle measure in the polygon?

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

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

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

Cheers,
Brent