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The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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Question Stats: 66% (02:27) correct 34% (02:27) wrong based on 1677 sessions

### HideShow timer Statistics The figure shown above consists of a shaded 9-sided polygon and 9 unshaded isosceles triangles.For each isosceles triangle,the longest side is a side of the shaded polygon and the two sides of equal length are extensions of the two adjacent sides of the shaded polygon.What is the value of a ?

A. 100
B. 105
C. 110
D. 115
E. 120

OG 2019 PS00947

Attachment: 11qk4nq.jpg [ 4.72 KiB | Viewed 34390 times ]

Originally posted by AbdurRakib on 16 Jun 2017, 07:01.
Last edited by bb on 01 Oct 2018, 09:26, edited 2 times in total.
Edited the question.
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The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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AbdurRakib wrote: The figure shown above consists of a shaded 9-sided polygon and 9 unshaded isosceles triangles.For each isosceles triangle,the longest side is a side of the shaded polygon and the two sides of equal length are extensions of the two adjacent sides of the shaded polygon.What is the value of a ?

A. 100
B. 105
C. 110
D. 115
E. 120

Attachment:
11qk4nq.jpg

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

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Originally posted by BrentGMATPrepNow on 27 Jun 2017, 14:06.
Last edited by BrentGMATPrepNow on 30 Apr 2020, 12:38, edited 2 times in total.
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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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)
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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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?
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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AR15J wrote:
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.

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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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pushpitkc wrote:
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 angles in a straight line are 190degree, b+140 = 180.
Thus b = c = 40degree

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

Thanks for the great breakdown!

(I think you meant "Since angles in a straight line are 180 degrees")
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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pushpitkc wrote:
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 angles in a straight line are 190degree, b+140 = 180.
Thus b = c = 40degree

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

Thanks for the great breakdown!

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

Yeah I meant that. Thanks for pointing out. Have corrected it.
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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pushpitkc wrote:
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)

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! Senior PS Moderator V
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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Hey dave13

Attachment: image.png [ 16.93 KiB | Viewed 27451 times ]

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

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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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pushpitkc wrote:
Hey dave13

Attachment:
image.png

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

pushpitkc Many thanks ! all is clear now! VP  D
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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pushpitkc wrote:
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)

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 Senior PS Moderator V
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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dave13 wrote:
pushpitkc wrote:
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)

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!
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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GMATPrepNow this was such a great/detailed explanation! Thank you!
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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pushpitkc wrote:
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)

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
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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Probus wrote:
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
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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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AbdurRakib wrote: The figure shown above consists of a shaded 9-sided polygon and 9 unshaded isosceles triangles.For each isosceles triangle,the longest side is a side of the shaded polygon and the two sides of equal length are extensions of the two adjacent sides of the shaded polygon.What is the value of a ?

A. 100
B. 105
C. 110
D. 115
E. 120

OG 2019 PS00947

Attachment:
The attachment 11qk4nq.jpg is no longer available

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
Attachments 11qk4nq.jpg [ 9.85 KiB | Viewed 6097 times ]

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Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un  [#permalink]

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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! Re: The figure shown above consists of a shaded 9 - sided polygon and 9 un   [#permalink] 28 Mar 2020, 18:08

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