The logo of a certain corporation is in the shape of a polyg

From figure.,

x + y = 180
6*x = 180
x = 30, y = 150

Let number of sides be "n", hence number of interior angles is also "n"
Sum of all interior angles = 150*n
Formula for sum of interior angles = 180*(n-2)

180*(n-2) = 150*n
30*n = 360

n = 360/30 = 12

Answer is D

While I would like to stay away from high level math and formulas, it helps to remember few formulas in case of Geometry questions. There are two pertinent flavors of the same formula that one should remember:


1 - sum of interior angles of a polygon with n sides = 180 * (n - 2)
2 - Measure of each angle of a regular polygon* = \(\frac{[180 * (n - 2) ]}{n}\)

A regular polygon* is a polygon that is equiangular* (all angles are equal in measure) and equilateral* (all sides have the same length)
* https://en.wikipedia.org/wiki/Regular_polygon

So in this question, we have 2 simultaneous equations:

y + x = 180
y - 5x = 0

Solving these equations we get, y = 150

Using #2

150 = \(\frac{[180 * (n - 2) ]}{n}\)[/color]

Solving this we get n = 12

How did you get that step, I'm really bad at geometry...

We are given y = 5x in the question statement. The two angles are supplementary (the two different angles made by the same line with another line) and hence add up to 180.

hi MacFauz,

x + y = 180
6*x = 180
x = 30, y = 150

i did this way 360/30 = 12 (since all angels are equal)
I guess this will not work out in all cases !

Also how u figured out this ?
Sum of all interior angles = 150*n

There are "n" sides and since the figure is a polygon, there are "n" angles. Each interior angle is 150. So sum of all interior angles is 150 added "n" times or 150*n

x+y = 180 (straight line)
x+5x= 180. x=30.

Interior angle of polygon = 30 x 5 = 150.

Interior Angle of regular polygon with side n no. of sides = (2n -4) * 90 / n
150 = (2n -4) * 90 / n. Solve it. n = 12.
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hi MacFauz,

x + y = 180
6*x = 180
x = 30, y = 150

i did this way 360/30 = 12 (since all angels are equal)
I guess this will not work out in all cases !



Basically what you have done(in red) is actually true for all regular polygons. We have the exterior angle as 30 degrees from the problem.Also, the sum of all the exterior angles in case of any polygon(regular/not regular) is always 360 degrees. As it is mentioned that this is a regualr polygon, only because of that, you can divide and get the answer as 12. Thus using the formula (n-2)*180 is not necessary, atleast in this problem.
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Hi vinaymimani.,

That was actually shanmugamgsn's solution.. Great if it is correct.. But I would have needed to used the (n-2)*180 formula since I did not know that sum of all exterior angles of a polygon is 360.. I have not heard of that before... Can you please illustrate the same in cases of a rectangle and triangle?

Incase of a rectangle, the external angle for each side is (180-90)degrees = 90 degrees. And there are 4 such angles. Thus the total sum of the external angles is 90*4 = 360 degrees.

Incase of a triangle, let the 3 angles be a,b,c. a+b+c = 180 degrees.Now, the respective exterior angles are : 180-a, 180-b,180-c. Thus, their sum is 180*3-(a+b+c) = 180*2 = 360 degrees.

I hope it's clear now. Once again, we can only divide 360 degrees by 30 degrees in this problem only because it is a regular polygon.
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Very simple. \(y=(n-2)180/n\). \(x=180-[(n-2)180/n]\). Given y=5x. Substitute and solve for n.

Hint: After equating 180 is cancelled throughout the equation. This makes calculations very easy.

Why does this apply? Why n-2? I understand that the degrees of x + y = 180, but what relevance does that have in regards to the above 180*(n-2)?

I don't understand.

For a polygon with 'n' sides, the sum of the interior angles is 180*(n-2)

For example, a triangle has 3 sides. So the sum of the interior angles is 180*(3-2) = 180


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Can you tell me why

Sum of all interior angles = 150*n?

Are all the angles equal?

Hi thegame001.,

The question states that all the angles of the polygon are equal.

Since there are n sides and hence 'n' angles, the sum of all the angles is 150 added 'n' times and that is 150n.


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Here's a quicker way.

y + x = 180
y = 5x
Hence, x = 30

Exterior angle, x = 360/n (since all angles are equal)
n= 360/30
n = 12

Hope this helps.

Hi All,

One of the rarer rules that can appear on the GMAT is "add a side, add 180 degrees"

For example
A triangle = 180 degrees
A 4-sided shape = 360 degrees
A 5-sided shape = 540 degrees
A 6-sided shape = 720 degrees
Etc.

In this prompt, we can use a combination of Geometry rules and the Answer choices to TEST THE ANSWERS.

To start, we have to figure out the values of X and Y. Since those two angles form a line (thus, X+Y = 180) and Y = 5X, then X+5X = 180.... X=30 and Y=150

We're told that each of the angles in this unknown polygon is the SAME. We're asked for the number of sides that this polygon has.

Using the above math rule, and the answer choices, we can work our way through the list until we find the polygon with equal 150 degree angles. Let's start with Answer B...

Answer B: 6-sided

A 6-sided shape = 720 degrees
720/(6 sides) = 120 degrees per side
This is TOO SMALL (it's supposed to be 150 degrees), so we clearly need more sides/more degrees.

Answer D: 12-sided

A 12-sided shape = 180 + 9(180) = (10)180 = 1800 degrees
1800/(12 sides) = 150 degrees per side
This is an exact MATCH for what we were told, so this must be the answer.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

sum of interior angles = \(180(n-2)\)

number of sides = \(\frac{180(n-2) }{ n}\)

From the question we create the below equation

\(150 = \frac{180(n-2) }{ n}\)

\(150 = \frac{180n - 360 }{ n}\)

\(150n = 180n - 360\)

\(30n = 360\)

\(n = 12\)
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