Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

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

Show Tags

16 Jun 2017, 07:01

3

This post received KUDOS

Top Contributor

25

This post was BOOKMARKED

00:00

A

B

C

D

E

Difficulty:

65% (hard)

Question Stats:

65% (01:54) correct 35% (01:56) wrong based on 555 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 ?

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

Show Tags

16 Jun 2017, 07:24

6

This post received KUDOS

2

This post was BOOKMARKED

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) _________________

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

Show Tags

25 Jun 2017, 09:50

2

This post received KUDOS

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.

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 ?

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

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

Show Tags

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")
_________________

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

Show Tags

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")

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

Show Tags

07 Jan 2018, 10:54

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)

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)

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)