The measures of the interior angles in a polygon are consecutive integers. The smallest angle measures 136 degrees. How

[BTGmoderatorDC]

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


OA B

Source: Manhattan Prep

[Jay@ManhattanReview]

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

OA B

Source: Manhattan Prep

Note that for a polygon, the sum of exterior angles is 360º. Thus, given that the smallest interior angle is 136º, the greatest exterior angle = 180 – 136 = 44º.

Since the angles are consecutive integers, the exterior angles would be 44, 43, 42, 41, ... & [44 – (n – 1)].

Or, the last ext. angle would be [44 – (n – 1)] = (45 – n)º

The sum of ext. angles = n * average of ext. angles = n*(Average of First angle & Last angles)

= n*[44 + (45 – n)] = n(89 – n)º

=> n(89 – n) = 360

Now instead of solving this quadratic equation, let's do plug-in value. We see that only n = 9 works.

The correct answer: B

Hope this helps!

-Jay
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[Scott@TargetTestPrep]

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


OA B

Solution:

Let’s assume that the polygon has n sides. Recall that the sum of the angle measures of an n-sided polygon is 180(n - 2). We are given the smallest angle of the polygon measures 136 degrees, so ones after that are 136 + 1, 136 + 2, and so on, ending with the largest angle measure of 136 + n - 1. Therefore, we can create the equation:

136 + (136 + 1) + (136 + 2) + … + (136 + n - 1) = 180(n - 2)

(135 + 1) + (135 + 2) + (135 + 3) + … + (135 + n) = 180n - 360

On the left hand side of the equation, we can rearrange the terms as:

(135 + 135 + 135 + … + 135) + (1 + 2 + 3 + … + n) = 180n - 360

There are a total of n repetitions of 135, so the sum of the 135’s is 135n; thus, we have:

135n + (1 + 2 + 3 + … + n) = 180n - 360

1 + 2 + 3 + … + n = 45n - 360

We use the shortcut formula for the sum of n consecutive integers to re-express the left side of the equation:

n(n + 1)/2 = 45n - 360

n^2 + n = 90n - 720

n^2 - 89n + 720 = 0

(n - 80)(n - 9) = 0

n = 80 or n = 9

Since only 9 is given, the correct choice is B.

Alternate Solution:

Recall that the total degree measure of an n-sided polygon is 180(n - 2), and thus the average degree measure (per angle) is 180(n - 2) / n. Now let’s check the answer choices if you don’t want to solve it algebraically.

A) 8

If the polygon has 8 sides, then the average angle measure is 180(8 - 2)/8 = 45(6)/2 = 135 degrees. However, we are given that the smallest angle has a degree measure of 136 degrees, choice A can’t be the correct answer.

B) 9

If the polygon has 9 sides, then the average angle measure is 180(9 - 2)/8 = 20(7) = 140 degrees. We are given that the smallest angle has a degree measure of 136 degrees and the degree measure of the angles are consecutive integers; thus the largest angle would have a measure of 136 + 8 = 144 degrees (if the polygon has 9 sides). Since the average of the smallest angle and the largest angle is (136 + 144)/2 = 280/2 = 140 degrees, exactly the same as the average of all angles, choice B must be the correct answer.

Answer: B