Sum of Interior Angles of a polygon is 180(n-2) where n is the number of sides (so is the number of angles);

We are told that the smallest angle is 136 degrees, the next will be 136+1 degrees, ..., and the largest one, n_{th} angle, will be 136+(n-1) degrees. The sum of the n consecutive integers (the sum of n angles) is given by \frac{first+last}{2}*# \ of \ terms=\frac{136+(136+n-1)}{2}*n=\frac{271+n}{2}*n;

So we have that 180(n-2)=\frac{271+n}{2}*n --> 360(n-2)=(271+n)*n, now try the answer choices: in order RHS to end with zero (as LHS is because of 360) then n, out of the options listed, could be either 10 or 9, n=9 fits.

Answer: B.
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Karishma,
Can you explain in more details when you say "The average of the given angles can only match 140 (such that effectively, all the angles are 140)". I did not get your point. Thanks
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Ajit

Sum of interior angles of a polygon = (n-2)*180 (not necessarily regular polygon)

Why? See the figure below:


A 6 sided polygon can be split into 4 triangles each of which has a sum of interior angles 180 degrees.
An n sided polygon can be split into n - 2 triangles.

When the polygon is regular, each angle is same so the sum is divided by the number of sides to get the measure of each angle e.g. in a regular hexagon, each interior angle = 4*180/6 = 120 degrees.

Now if I have a hexagon whose angles are 115, 117, 119, 121, 123 and x, what will be the angle x?
We can see it in two ways -
1. The sum of all angles should be 4*180 = 720
So 115 + 117 + 119 + 121 + 123 + x = 720
or x = 125


2. The average of the angles should be 120. (Since the sum of the angles is 720 and there are 6 sides)
119 and 121 average out as 120. (119 is 1 less than 120 and 121 is 1 more than 120)
117 and 123 average out as 120.
So 115 and x should average out as 120 too. Therefore, x should be 125.

In the question, the average of the given angles of the polygon can only be 140. So it must have 9 sides. To confirm,
136, 137, 138, 139, 140, 141, 142, 143, 144 - 9 angles with average 140. So the polygon must have 9 sides.

(It cannot be 144 or anything else because 10 angles (136, 137, 138, 139, 140, 141, 142, 143, 144, 145) will not give an average of 144)
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Karishma
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Actually, I wouldn't say it is very tough if we know that the sum of interior angles of a polygon is 180(n -2) (I explained above why it is so). We can also use a very straight forward but long approach.

Interior angles of the given polygon: 136, 137, 138, 139, 140, 141, 142, 143....

Using options:

If the polygon had 8 sides, it would have had 8 interior angles. Their sum: (8-2)180 = 1080
Sum of 8 angles: 136 +137 + 138 + 139 + 140 + 141 + 142 + 143 is more than 1080 hence this polygon does not have 8 sides.

If the polygon had 9 sides, it would have had 9 interior angles. Their sum: (9-2)180 = 1260
Sum of 9 angles: 136 +137 + 138 + 139 + 140 + 141 + 142 + 143 + 144 is 1260. Hence this polygon does have 9 sides.
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ye... well i guess i need to work on my geometry more so the way i process these things will be better...

thanks both as usual.
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