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

Question

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°

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

anairamitch1804 · Accepted Answer

Posting OE of this problem.

gracie · Answer

140n=180(n-2)
n=9
smallest interior angle=140°-4°=136°
C

JeffTargetTestPrep · Answer

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

rishit924 · Answer

Dear Mam,

I was in an impression that 180(n-2)/n equation is applicable for regular polygons only.
Sum of angles we can find for any but divide by n is only possible for regular polygons.

Please advise.
Thanks in advance.

IanStewart · Answer

180(n-2) is the sum of the n angles of any n-sided polygon. When we divide that by n, the total number of angles, we are finding the average size of an angle in an n-sided polygon. If the polygon is regular, every angle is the same, so in that case, every angle is equal to the "average" angle.

In a non-regular polygon, you can still calculate the average angle in your shape in this way. The average of the four angles in any quadrilateral, for example, will always be 90 degrees, even if no angle equals 90 degrees. In this question, because the angles are consecutive integers, the average and the median are equal (that's always true in any equally spaced set). So in this question, the average angle equals the median angle, so 180(n-2)/n = 140, and from there you can find n and answer the question.

Archit3110 · Answer

the median of the angle sum can be written as
140 = 180(n-2)/ n
solve for n = 9
all consecutive angles 
136,137,138,139..... 144
smallest is 136
option C

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

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