dabaobao wrote:
The measures of the interior angles in a polygon are consecutive odd integers. The largest angle measures 153 degrees. How many sides does this polygon have?
A) 8
B) 9
C) 10
D) 11
E) 12
Solution: Using Sequence
Sum of Interior angles = (n-2) * 180
Max Angle = 153; Min Angle = 153 - 2(n-1)
Since we have an evenly spaced set, sum of interior angles = ((a1 + an)/2)*n
((153 - 2(n-1) + 153)/2)*n = (n-2) * 180
(153 -n + 1)*n = 180n - 360 => 153n -n^2 +n = 180n - 360
=> n^2 + 26n - 360 = 0 => (n + 36) (n - 10) = 0 => n = 10
ANSWER: C
Veritas Prep Official Solution
Angles of the polygon: 153, 151, 149, 147, 145, 143, 141, … , (153 – 2(n-1))
The average of these angles must be equal to the measure of each interior angle of a regular polygon with n sides since the sum of all angles is the same in both the cases.
Measure of each interior angle of n sided regular polygon = Sum of all angles / n = (n−2)∗180n
Using the options:
Measure of each interior angle of 8 sided regular polygon = 180*6/8 = 135 degrees
Measure of each interior angle of 9 sided regular polygon = 180*7/9 = 140 degrees
Measure of each interior angle of 10 sided regular polygon = 180*8/10 = 144 degrees
Measure of each interior angle of 11 sided regular polygon = 180*9/11 = 147 degrees apprx
and so on…
Notice that the average of the given angles can be 144 if there are 10 angles.
The average cannot be higher than 144 i.e. 147 since that will give us only 7 sides (153, 151, 149, 147, 145, 143, 141 – the average is 147 is this case). But the regular polygon with interior angle measure of 147 has 11 sides. Similarly, the average cannot be less than 144 i.e. 140 either because that will give us many more sides than the required 9.
Hence, the polygon must have 10 sides.
Answer (C).
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