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

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MBAB123

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19 Dec 2020, 03:02

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basshead

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.

basshead, how do you know that all the interior angles of the polygon are equal?

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28 Dec 2020, 14:56

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Video solution from Quant Reasoning:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1

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10 Jan 2021, 12:13

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AbdurRakib

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

Show Spoiler

Attachment:

11qk4nq.jpg

Of the triangle, the largest side is the shaded side of the polygon (given) so the other two sides are equal, with equal angles. Each angle is transversal to the other angle of the other triangle. Let the other angles are x each. So, every internal angle of the polygon will be 180-x.

The angles of the polygon =180(9-2)=180*7=1260, each internal angle is =1260/9=140, Thus, 180-x=140, x=40, Henceforth a=100,

The answer is A

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17 Mar 2021, 16:14

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To Alexpol;
Since all the triangles are isosceles and the longest sides of all the triangles are equal, then all the interior angles of the shaded regions are also equal. That by extension makes it a regular polygon.

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It is a lot easier to consider the exterior angles in this question. We know that sum of exterior angles is 360 for a polygon.

From the explainations above we know that each exterior angles is equal and there are a total of 9 exterior angles.

so --> 9 * (Exterior angle) = 360
Exterior angle = 40

Now to find a we know that sum of angles in a triangle is 180 and since it is a isoceles triangle:

exterior angle + exterior angle + a = 180
40 + 40 + a = 180
a = 100

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31 Mar 2022, 07:00

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Question: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

GMAT Geometry questions can sometimes be tricky or at least they "appear" to be complicated like this one here.

But with a little practice, you can master these questions in no time.

In this post, we will discuss the properties of regular polygons and solve the official question asked here.

GMAT Geometry-Concept on Polygons:

Polygons are figures with straight sides and angles. The most common type of polygon is a triangle, which has three sides and three angles. Quadrilaterals are polygons with four sides and four angles. Polygons can have any number of sides, but the more sides a polygon has, the more complicated the question "appears".
A regular polygon is a polygon that has all sides and angles equal or else its an irregular polygon.

GMAT questions about polygons can ask about the properties of specific types of polygons, or about the relationships between the angles and sides of polygons.
Some common properties that are tested include:

The sum of the angles in a n-sided polygon is (n-2)*180 degrees and if its a regular polygon then each internal angle = (n-2)*180 / n degrees

The interior angles of a polygon add up to 360 degrees and if its a regular polygon then each external angle = 360/n degrees

Number of diagonals in a n sided polygon = n *(n-3)/2

Now, lets get back to our question.

Pls also refer to the attached figure as you scroll down to understand the solution.

In the triangle ABC, side of the shaded polygon is BC, it is the longest side in the triangle and hence the other two sides that is AB & AC are the equal sides.

GMAT Track of thought 1- Angles opposite to equal sides in a triangle are equal.

So angle ABC = angle ACB. For simplicity, lets call it x.
What now? How do we proceed from here?

GMAT Track of thought 2 - Vertically opposite angles created by 2 intersecting lines are equal!

This implies angle DBE and angle GCF are also each equal to x degrees.
Since all the triangles are mentioned to be isosceles, let us apply the concepts of Angles opposite to equal sides in a triangle and Vertically opposite angles created by 2 intersecting lines to be creating equal angles.
Thus, all angles marked in black in the figure are equal.

GMAT Track of thought 3-For a pair of linear angles,if one angle is x,the other is 180-x.

I have used this concept and marked the linear pairs corresponding to the angles marked in black (in white).
They are all equal!!
What does this lead us to??
It says a lot of the polygon shaded in grey.
All angles marked in white are the internal angles of the polygon and are equal.
Hence,the polygon is a regular polygon!

Using the concept that in a regular polygon each internal angle = (n-2)*180 / n degrees and substituting n=9,we have

Each internal angle (9-2) * 180 / 9 = 140 degrees. So all the angles marked in white are 140 degrees

This implies 180-x = 140 =>x (angles in black) = 40 degree

Thus the equal angles in the triangle ABC are 40 degree each

Hence the unequal angle or 'a'= 180- (40+40)

= 180-80

=100 degrees

(option a)

Do you still think this question should be a 700 or was this figure an illusion?

Let me know in the comments below.

Devmitra Sen
GMAT Mentor

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31 Mar 2022, 07:10

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But with a little practice, you can master these questions in no time.

In this post, we will discuss the properties of regular polygons and solve the official question asked here.

GMAT Track of thought 1- Angles opposite to equal sides in a triangle are equal.

So angle ABC = angle ACB. For simplicity, lets call it x.
What now? How do we proceed from here?

Let me know in the comments below.

Devmitra Sen
GMAT Mentor

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05 Oct 2022, 01:57

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AbdurRakib

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Attachment:

11qk4nq.jpg

Can't we solve this question using formula of sum of all external angles of polygon = 360.
Since all the equal angles of the triangle are the external angles on the polygon?