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# In the figure shown, what is the value of v+x+y+z+w?

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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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01 Jul 2014, 04:17
Attachment:

star-sol.png [ 10.08 KiB | Viewed 2615 times ]

i didn't know the central angle theorem, and tried solving with a different way.
sorry for my mad paint skills :D

here is my solution:
1. Draw a line via vertex of angle Y, parallel to the line between angles V and Z. In a picture the red coloured lines are parallel.
2. Draw a line via vertex of angle Y, parallel to the line between angles X and Z (violet coloured)
3. Draw a line via vertex of angle Y, parallel to the line between angles X and W (blue coloured)

The following can be concluded from the pic accoridng to thales theorem:
a. angles between red and violet lines will be same (angle Z)
b. angles between black and red lines will be same (angle V)
c. angles between blue and black lines will be same (angle W)
d. angles between blue and violet lines will be same (angle X)

as a result sum of 5 angles will be a violet line and equal to 180 degrees

My solution is obviously not as simple and quick as Bunuel's one, but maybe you can use my approach for solving similar problems

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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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19 Nov 2014, 08:54
Bunuel wrote:
Let's simplify the problem by imagining that we have a star that is inscribed in a circle as shown below:

Hi Bunuel. How can we assume that the star could be inscribed within a circle in the absence of information. Further, the general GMAT assumption is that all diagrams are not drawn to scale unless the contrary is mentioned.

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JA
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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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19 Nov 2014, 10:21
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Wow! See what I found with a little googling!

A star is always regularly shaped (this clarifies why Bunuel assumed that the start would get inscribed in the circle)!
1. The sum of the angles formed at the tips of the five pointed star is 180; the sum of the angles formed at the tips of the six pointed star is 360.
2. The formula for the sum of the angle measurements at the tips of an n-pointed star is f(n)=180(n)-720 where n is an integer greater than 4.

This clarifies my doubts Very Happy
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JA
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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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15 Dec 2014, 22:57
Hi Bunuel,
can you please elaborate how you made the assumption that "Let's simplify the problem by imagining that we have a star that is inscribed in a circle as shown below"?

Thanks alot.

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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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27 Aug 2015, 17:08
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Interior angle of a polygon = $$((n-2)180)/n$$
For Pentagon = $$(5-3)180/5$$ = 108
sum of angles in each triangle angle x+(180-108)+(180-108) = 180 .... x = 36
there are 5 triangles ... 36*5 = 180

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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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16 Mar 2016, 06:48
Alternate Solution with just the basics.
(I have not named every vertex separately in order to avoid confusion (mess). Every vertex of the star is named by the angle it depicts in the picture)
Attachments

File comment: y+A+B+C = 360 (sum of all the angles in quadrilateral-YABC )
A= 180-(y+v) (sum of all angles in a triangle is 180,Triangle YAV)
B= 180-(x+z) (Triangle XBZ)
C= 180-(y+w) (Triangle YCW)

Now, Substitute the values of A, B and C in the equation : A+B+C+y=360
(180-y-v) + (180-x-z) + (180-y-w) + y =360
By solving the above, we get:
540 - w - v - z - x - 2y + y=360
x+y+v+z+w=180

Govind Kohli
Kudos if this helps

step0004.png [ 5.55 KiB | Viewed 1722 times ]

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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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01 Apr 2016, 02:05
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$$A + B + C + D + E = (5-2)*180 = 540$$

$$A+x+z=180$$
$$B+y+v=180$$
$$C+x+w=180$$
$$D+v+z=180$$
$$E+y+w=180$$

$$A + B + C + D + E +2x+2y+2z+2w+2v =900$$

$$540 + 2x+2y+2z+2w+2v = 900$$

$$2x+2y+2z+2w+2v = 360$$

$$x+y+z+w+v=180$$
Attachments

GMAT.png [ 8.59 KiB | Viewed 1803 times ]

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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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05 Nov 2016, 14:29
I think the easiest way to approach this problem was assuming we were dealing with a regular pentagon (all angles equal) and figuring out what each angle was using the interior angle formula --> (5-1)(180) = 540 --> 540/# of angles = 108

We know that the angle outside, opposite the interior angle of the pentagon is the same, therefore the two adjacent triangle angles will be 360-216 = 144. Divide by 2 to get them symmetrical and you will find each triangle off the pentagon has two angles that are 72 degrees, making the third of each 36.

36x5=180

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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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07 Nov 2016, 02:10
pentagon sum of angles is 540 degrees. Every given angles is approximately 1/3 of pentagon angle. So, sum is (1/3)*540=180

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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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23 Jan 2017, 12:47
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Bunuel wrote:
The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

In the figure shown, what is the value of v+x+y+z+w?

(A) 45
(B) 90
(C) 180
(D) 270
(E) 360

First, let's consider a PERFECT star.

Notice that the pentagon in the center is a perfect (regular) pentagon, which means ALL 5 angles are equal.
The sum of the angles in an n-sided figure = 180(n-2) degrees
So, the sum of the angles in this 5-sided figure = 180(5-2) = 180(3) = 540 degrees
Since ALL 5 angles are equal, then the measure of each angle = 540/5 = 108 degrees.

Since two angles on a line must add to 180 degrees, we can see that the angles adjacent to the 108-degree angles must equal 72 degrees (since 180 - 108 = 72)

At this point, we can see that we're dealing with 5 triangles, and for each triangle, we know two of the angle measurements.
Since the sum of the angles in a triangle = 180, we know that each missing angle = 36 degrees (180 - 72 - 72 = 36)

So, v + x + y + z + w = 36 + 36 + 36 + 36 + 36
= 180 degrees

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In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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30 Jul 2017, 03:44
Although GMAT pictures may be not completely accurate and drawn to scale, I realize they don't usually try to "trick" users... Therefore I just drew the figure on a piece of paper and given the answer choices I realized the sum of the angles must be somewhere around 180 degrees:
Attachment:

star.png [ 31.16 KiB | Viewed 841 times ]

Probably this approach is not the most scientific and reliable way to solve the problem, but I guess it worked on this one.

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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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25 Nov 2017, 09:05
In a regular pentagram (5-pointed star, NOT Pentagon), the angle in each point is 36 degrees, so the angles in all five points sum to 180 degrees.
Attachment:

1.jpg [ 8.1 KiB | Viewed 311 times ]

In an irregular pentagram (NOT Pentagon), the angles might be all different from each other, but the angles in all five points still sum to 180 degrees.
Attachment:

2.jpg [ 17.66 KiB | Viewed 312 times ]

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Re: In the figure shown, what is the value of v+x+y+z+w?   [#permalink] 25 Nov 2017, 09:05

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