The measures of the interior angles in a polygon are consecutive integ

Just another way of doing this sum:
The sum of exterior angles for any polygon = 360 degrees.Now, given that the minimum internal angle measure is 136 degrees--> the exterior angle = 180-136 = 44 degrees.
Also, we know that this value will keep decreasing like 43,42,41 etc. It is easy to see that only if there 9 terms, the middle value is 40 and we know that 40*9 =360 degrees.

B.

When I look at this question, I say, "I know how to find the Interior angle of a regular polygon. But this is not a regular polygon since it has angles 136, 137, 138, 139, 140, 141, 142 ....etc."

Also,
Interior angle of 8 sided regular polygon = 180*6/8 = 135
Interior angle of 9 sided regular polygon = 180*7/9 = 140
Interior angle of 10 sided regular polygon = 180*8/10 = 144
etc

The average of the given angles can only match 140 (such that effectively, all the angles are 140) Hence, the polygon must have 9 sides.

Remember, capitalize on what you know.

Sum of interior angles for n sides: (n-2)*180
And each interior angle are increasing by 1.
This can be written as : 136+137+138+139+... = 136+(136+1)+(136+2)+(136+3)+...
For n sides

136n+(1+2+3+4....+n-1)
136n+(n-1)n/2 --- Sum of n-1 natural number

Equate:
136n+(n-1)n/2 = (n-2)*180
n^2-89n+720=0
n^2-80n-9n+720=0
n(n-80)-9(n-80)=0
(n-9)(n-80)=0

n=9
n=80

Since 9 is one of the options.

Ans: "B"

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)

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.

Another way of solving the problem

sum of internal angles of a polygon = (n-2)*180. as sum is multiplied by 180, the unit degit of sum should be 0.

also looking at the options least number of sides is 8.

sum of unit degits for 8 sides is 6 (from 136) + 7+8+9+0+1+2+3 = 6.
sum of unit degist for 9 sides is 6+7+8+9+0+1+2+3+4 = 0.
Hnec option B is correct anser

Remember that sum of angles in a polygon=(n-2)180. This means that whatever the value of n there will always be a 0 in units digit. Now, the question becomes 'How many consecutive integers we should add starting from 136 so that units digit is 0. 6+7+8+9+0=(3)0. Note that 3 is put in brackets bcos it is carried over. Now we know that sum of angles in a pentagon=540. But does 136+137+....140 equal 540. Use formula \(\frac{n}{2}\)(a+l)=\(\frac{5}{2}\)*(136+140)=690. Does not match. So proceed further. 6+7+8+9+0+1+2+3+4=(4)0. Now the series becomes 136+137+138+........+144 consisting of (144-136+1=)9 elements. Again use above formula for sum and it comes out to be 1260. Now we know that sum of angles in a nonagon is (n-2)180=1260. This choice matches. So, answer is 9.

Hey guys...please let me know whether this method is correct or not.
the stem says that the angles are consecutive integers, 136 being the smallest one. Also going by the formula of sum of interior angles, we know atleast this much that sum of angles cant be number like 729, 847, 653, 542 but it can be a number of the form xx0. So going by this knowledge, if the smallest angle is 136 then the sum of angles can only be of the form xx0 when the largest angle is 144. That is 9 sides.
Yeah the largest angle could be 154, 164 but in that case the number of sides must be 19 and 29 respectively.
Since the largest option is 13, hence the answer is 9.
Please correct me if I am wrong.

Reverse Approach using external angle

We are given : smallest internal angle = 136 deg ;

to find : number of sides 'n'

Solution : We know that, internal angle = (n - 2)*180;

But, we dont know value of n. However, we know one thing for sure. Irrespective of the value of n, Sum of all the external angles will be 360 deg.

So, Corresponding external angle for internal angle of 136 deg = 180 - 136 = 44 deg. (Since, Sum of internal + external angle = 180 deg)

As internal angle increases by 1 external angle decreases by 1.

So, now 2nd external angle will be 43 deg, 3rd external angle will be 42 deg, 4th will be 41 deg and so on. We keep doing this till the point our sum of all external angles turns out to be 360 deg.

So, 44 + 43 + 42 + 41 + 40 + 39 + 38 + 37 + 36 = 360

total number of terms in above equation is 9. So the number of sides of polygon = 9

( We can also use concept of AP. All the terms are in AP. We know S = 360, t1 = 44, d = -1, n=?
360 = n/2* (2*44 + (n-1)*-1) => n^2 - 89n + 720 = 0 => n = 80 or n = 9)

sum of angles =180(n-2) where n is the number of sides
since we have consecutive angles, then median=mean

B n=9 ,then the sum of angles =180*7

the median of 9 consecutive integers is 140 (136 137 138 139 140 141 142 143 144)
the sum of consecutive integers is 140*9

180*7/140*9 =(20*9*7)/(20*7*9)= 1

OFFICIAL SOLUTION

We are told that the smallest angle measures 136 degrees--this is the first term in the consecutive set. If the polygon has S sides, then the largest angle--the last term in the consecutive set--will be (S - 1) more than 136 degrees.

The sum of consecutive integers = (Average Term) * (# of Term)= \(\frac{First + Last}{2}\) * (# of Terms).

Given that there are S terms in the set, we can plug in for the first and last term as follows:

\(\frac{136 + 136 + (S-1)}{2} * S\) = sum of the angles in the polygon.

We also know that the sum of the angles in a polygon = 180 (S-2) where S represents the number of sides.

Therefore: \(180(S-2) = \frac{136+136+(S-1)}{2} * S\). We can solve

for S by cross-multiplying and simplifying as follows:

\(2(180)(S-2) = [272 + (S-1)] S\)
\(360S - 720 = (271 + S)S\)
\(360S - 720 = 271S + S^2\)
\(S^2 - 89S + 720 = 0\)

A look at the answer choices tells you to try (S - 8), (S - 9), or (S - 10) in factoring.

As it turns out (S-9)(S-80)=0, which means S can be 9 or 80. However S cannot be 80 because this creates a polygon with angles greater than 180.

Therefore S equals 9; there are 9 sides in the polygon.

The correct answer is B.

Why use quadratic equation when we can simply look at unit digit and options to answer this question

Check video solution here.

Answer: Option B

Whenever we talk of a polygon in GMAT, it is a convex polygon.

Their solution seems to miss the point. It's not inherently problematic that some angles in the hypothetical 80-side polygon are greater than 180 degrees. What is a problem is that one angle in such a polygon would need to be exactly 180 degrees, and you can't have a 180-degree angle in a polygon, because then you have a straight line and do not have a vertex.

There's nothing in the GMAT instructions that tells you to assume all "polygons" are convex (the Math Review section in the OG says something to that effect, but that Math Review is not a substitute for the test directions), but as a practical matter, if you did assume that when a GMAT question mentions a "polygon" that it is a "convex polygon", I think you'll always get the right answer -- I can't imagine seeing a real GMAT question about polygons where the answer would be different if you considered concave polygons separately from convex ones. If you see a polygon with interior angles greater than 180 degrees on the GMAT, the question will include a diagram so you can see what kind of polygon you're dealing with. Q10 in the diagnostic test at the start of the OG is one example (the star shape is concave if you erase the edges of the pentagon).

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

tough one. simple solution, but its not an ez one.
am i right?

I wouldn't say it's very hard either, though I wouldn't recommend trial and error in this cases:

...
Calculating the sum of the interior angles of a polygon with 7 sides then checking whether 7 consecutive integers starting from 136 add up to that value;
Calculating the sum of the interior angles of a polygon with 8 sides then checking whether 8 consecutive integers starting from 136 add up to that value;
Calculating the sum of the interior angles of a polygon with 9 sides then checking whether 9 consecutive integers starting from 136 add up to that value;

Whereas equating two formulas and working on answer choices should give an answer in less time: \(180(n-2)=\frac{271+n}{2}*n\) --> \(360(n-2)=(271+n)*n\) --> \(n=9\).