The figure shown above consists of a shaded 9 - sided polygon and 9 un

Although the 9-sided polygon is NOT a regular polygon (with equal angles and equal sides), we CAN show that the angles are all equal.
The key here is that each of the unshaded triangles is an isosceles triangle.

So, for example, we know that the two angles with the red dots are equal.


Since the next angle over is opposite the angle with the red dot, we know that that angle is also equal to the other two angles (see the diagram below)


Since the next triangle is ISOSCELES, we know that its other angle is equal to the first (see the diagram below)


We can continue applying the same rules and logic to see that all of the red dotted angles are all equal.


Next, if all of the red-dotted angles are equal, then all of the angles denoted with x are all equal, since each of those angles is on a line with the red-dotted angle.


To determine the value of x, we'll use the following rule:
the sum of the angles in an n-sided polygon = (n - 2)(180º)
So, in a 9-sided polygon, the sum of ALL 9 angles = (9 - 2)(180º) = (7)(180º)
So, EACH individual angle = (7)(180º)/9 = 140º
Also, since the red-dotted angle is on the line with the 140º, the red-dotted angle = 40º
Let's add this information to the diagram below:



Since the unshaded triangle is isosceles, the other angle is also 40º


Since all three angles in the triangle must add to 180º, the last remaining angle (aº) must be 100º

Answer:

RELATED VIDEOS

In an n-sided polygon the sum of internal angles is (n-2)*180 degree
So, the 9-sided polygon has sum of internal angles as 7*180 degree,
making each of the individual angles \(\frac{7*180}{9}\) = 140 degree

Lets name the other two angles of the triangle with angle a as b and c.
Since we know that the triangles(isosceles) has two sides of equal length as extensions,
we know that

1. b=c (angles opposite equal sides are equal in magnitude)
2. a+b+c =180 degree (sum of angles in a triangle is 180 degree)

Since the sum of the angles in a straight line is 180 degree, b+140 = 180.
Thus b = c = 40 degree

Using this information in a+b+c =180 degree, we can deduce that a = 100(Option A)

In the question, it is not mentioned that this polygon is a regular polygon, how can we calculate the individual angle of 9-sided polygon?

Since the triangles are isosceles, two of their angles will be equal. Now, as both of those equal angles are supplements of the vertices of the nonagon (9-sided polygon), and one vertex of the nonagon is a common supplement of any two adjacent triangles, all the angles of the nonagon will be equal.

Hope it helps

Thanks for the great breakdown!

(I think you meant "Since angles in a straight line are 180 degrees")

Hi Hadrienlbb

Yeah I meant that. Thanks for pointing out. Have corrected it.

Hi pushpitkc,

can you please explain, how did you figure out 140 belonged to which angle? i did exatly the same and when i got 140 i thought "a" must be 140

thanks!

Hey dave13



In the 9-sided polygon, each and every internal angle is equal to 140 degree.
Using the property that sum of angles in a straight line equal 180 degree,
we can deduce that one of the angles in the triangle is equal to 40 degree.
Lets call that c, which is equal to 40 degree

We know that the sum of angles in a triangle is 180 degree.
a + b + c = 180
Since angles opposite equal sides are equal in magnitude(b=c=40)

a + 80 =180
a = 100

Hope this help you!

pushpitkc Many thanks ! all is clear now!

pushpitkc , one thing i dont get if penthagon has following properties:

Interior angle equals 108 degrees
Exterior angle equals 72 degrees

and if it is an Isosceles traingle which means two angles are equal then i though that these two equal angles are of 72degrees each, which mean that \(a\) should be 36 degrees

please calibrate my reasoning

Hi dave13

It is important to understand that in an isosceles triangle, the angles
opposite to the equal sides are equal. If you observe the diagram, you
will see that the angles "b" and "c" are opposite to the sides which are
equal.

Hope this helps you!

GMATPrepNow this was such a great/detailed explanation! Thank you!

Hi GMATPrepNow,

I think what you have shared in the response is very important.

That if we know that a polygon is a regular polygon then can we say each angle is \(\frac{180(n-2)}{n}\).
The red dot and blue angle make 180 and all the red dots are equal, hence each blue angle must also be equal.

Though i need a little more help on classification of Polygons
a All angles and All sides equal Regular Polygon
B all sides equal ( Well does this exist)
c all angles and sides are not equal - irregular Polygon

Thanks
Probus
_________________

a) Yes, in a regular polygon, all sides are equal, and all angles are equal
b) We can have polygons where the sides all have the same length, but the angles are not all the same (e.g., rhombus).
c) If the sides are not equal and the angles are not equal, then we have an irregular polygon.

Cheers,
Brent

It may NOT or rather it IS not correct to take all angles equal without working on it as it is not given that the polygon is a REGULAR polygon

Two ways to do it..
(I) Sum of exterior angles is 360 in all polygons..
Here all exterior angles are x and there are 9 of them, so \(9*x=360...x=\frac{360}{9}=40\)
Take any isosceles triangle now, \(a+x+x=180...a+40+40=180...a=100\)
(II) All angles of Polygon are equal
Each angle 1 to 9 added with x gives 180..Now this means all angles are equal and will be \(\frac{(9-2)*180}{9}=140\)
So, \(x+140=180...x=40\), and \(a =180-2*40=100\)

A

_________________

Hello guys,

I am actually wondering whether the wording: 'the two sides of equal length are extensions of the two adjacent sides of the shaded polygon.' is wrong? Two adjacent sides should be the 2 sides together. But it looks like the 2 sides of equal length are the extensions of 2 sides which are not adjacent to each other: there is a side between them. Can someone help? Thank you!

Hi!

I still do not get why everyone decided that it is the right polygon and all 9 interior angles are equal.

Can anyone explain this? Thanks.

If you pay attention, you can get away from this question without having to deal with the awkward drawing.


Each Isosceles Triangle's 2 Equal Sides are formed by extensions of the Adjacent Sides of the 9-Sided Polygon.

Also, as you move from Isosceles Triangle - to - Isosceles Triangle, you will notice that the Vertically Opposite Angles created by the intersection of the extended sides will be Equal as well.

Since the Vertically Opposite Angles are equal and the 2 Angles inside the Isosceles Triangle are Equal -----> you can carry this fact across the entire outside of the Polygon and Infer that the 2 Equal Angles in every Isosceles Triangle have the same measure.

Rule: If the Each Exterior Angle taken by extending the Side of the Polygon is Equal, then Each Interior Angle will be Equal as well.

Although we can not say that this is a Regular Polygon, we know that each of the 9 Interior Angles of the Polygon must be Equal.

(1st) Taking the measure of 1 Exterior Angle 1 time at each Vertex of the 9 Sided Polygon, each Exterior Angle = the 2 Equal Angles of the Isosceles Triangle = (360 degrees) / (9 Sides) = 40 degrees

(2nd) Since the 2 Equal Angles of Each Isosceles Triangle are equal to = 40 degrees, the 3rd Angle A (across from the NON-Equal Side of the Isosceles Triangle) must equal:

A = 180 - (40 + 40) = 100 degrees

Answer -A-

Sum of the angles in a 9 sided polygon = (7)(180) = 1260

Each interior angle measures 1260 / 9 = 140 degrees

We know each triangle is isosceles. Let each base angle = x.

We know 140 + x = 180. Therefore x = 40.

180 - 2x = 100

Answer is A.
_________________