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

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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

[Reveal] Spoiler:
Attachment:
11qk4nq.jpg
11qk4nq.jpg [ 4.72 KiB | Viewed 4118 times ]
[Reveal] Spoiler: OA

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Last edited by Bunuel on 16 Jun 2017, 07:12, edited 1 time in total.
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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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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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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.

Hope it helps :)

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AbdurRakib wrote:
Image

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

[Reveal] Spoiler:
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.
Image

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)
Image

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

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

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.
Image

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:
Image


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

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

Answer:
[Reveal] Spoiler:
A


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

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New post 26 Nov 2017, 15:26
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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New post 29 Nov 2017, 11:54
Hadrienlbb 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 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")


Hi Hadrienlbb

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] 29 Nov 2017, 11:54
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