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

I was thinking of using the exterior angle property with 2 triangles

Angle 1 = X + W
Angle 2 = v + Y

angle 1 + angle 2 + Z = 180

then add those angles up its 180 degrees.

SOLUTION

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


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

Let's simplify the problem by imagining that we have a star that is inscribed in a circle as shown below:

As we can see, 5 arcs subtended by the inscribed angles (x, y, z, w, and v) make the whole circumference. Hence the sum of the corresponding central angles must be 360 degrees, which makes the sum of the inscribed angels 360/2=180 degrees (according to the Central Angle Theorem the measure of inscribed angle is always half the measure of the central angle).

Answer: C.


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Hi,

With reference to the attached diagram:
In triangle DFB, angle DFB = 180 - angle FDB - angle FBD
=180 - v - z

Similary, in other triangles,
angle AJC = 180 - y - w
angle EIB = 180 - x - z
angle AHD = 180 - v - y
angle EGC = 180 - x - w

Adding all, angle DFB + angle AJC + angle EIB + angle AHD +angle EGC = sum of interior angles of pentagon
=540 = 5*180 - 2(v+w+x+y+z)
so, v+w+x+y+z = 180

Answer (C)

Regards,

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

Answer: C

I can suggest two solutions:

Solution A

We can compute the sum of the angles from the five triangles created on the sides of the pentagon ABCDE. In those triangles, we have five pairs of congruent angles (see them marked by colored arcs in the attached drawing). Those angles are external angles for the pentagon and their sum is \(360^o\). See at the end of the post the justification for the fact that in every convex polygon, the sum of the external angles is \(360^o\).
Therefore, v + x + y + z + w = 5 ∙ 180 – 2 ∙ 360 = 900 – 720= 180.

Solution B

Since the question is a multiple choice one, we can assume that there is one correct answer and that that answer does not depend on the shape of the “star”. Assuming that the star can be inscribed in a circle, we can see that the requested sum of the angles is 360/2 = 180, because each angle is inscribed in the circle and the five corresponding arcs complete the circle.
Remark: If one of the answers would have been “It cannot be determined” or something similar, than this argument wouldn’t work.

Correct answer: C

Sum of the external angles for a convex polygon:

We know that the sum of the interior angles in a convex polygon with n sides (n being a positive integer greater than 2) is given by the formula:
(n – 2)∙180 = 180n – 360.
Each external angle is 180 – the corresponding interior angle. Therefore, the total sum of the exterior angles is 180n – (n – 2) ∙ 180 = 180n – 180n + 360 = 360.

Note: Convex polygons have the property that each of their angles is less than 180. All the polygons dealt with on GMAT are convex (triangle, quadrilateral, pentagon, hexagon,...) or are made up of convex polygons. In this question, the figure of the star, without the sides of the small convex pentagon, is an example of a non-convex decagon: it has 10 sides, and 5 angles which are greater, and 5, which are smaller than 180.

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The sum of all the angles of the pentagon in the middle is (5-3)180 = 540.
Each vertex of the pentagon has a angle of 108 degrees (on average).
Lets look at triangle AYW, since A is a vertex of the pentagon, the sum of y + w = 72.
In this same scenario z + v also = 72 and x equals 36 (72/2).

72 + 72 + 36 = 180

Nice! In other words, if we understand from the answers that the sum does not depend on the shape of the star, we can consider a regular pentagon. Then each of its angles is 108, and each angle of the star is 36, so the sum is 180.
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Hi Bunuel,

I didn't understand the part where you used "Central Angle Theorem". Is it correct to assume that if I see one or more triangles then If I draw a
circle around those triangles which are kind of inter-connected then can I use the "Central Angle Theorem"?

Thanks a lot for the help,

LR

Not sure understand what you mean.

Anyway, the Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle:


For more check here: math-circles-87957.html
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I'm ok with the Theorem, but I want to know how you came up the assumption " Let's simplify the problem by imagining that we have a star that is inscribed in a circle as shown below" as you mentioned above?

You are right, not always can we conclude that five points in a plan are co-circular; while it is true that three points are always co-circular.

However, the solution works if we imagine two circles for three points each. It was just a coincidence that the assumption that all the given points are co-circular worked.

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

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.

Read more here: https://mathforum.org/pom2/nov.98/winner.html

This clarifies my doubts Very Happy

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
Answer (C)

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)

\(A + B + C + D + E = (5-2)*180 = 540\)

\(A+x+z=180\)
\(B+y+v=180\)
\(C+x+w=180\)
\(D+v+z=180\)
\(E+y+w=180\)

When we add it all:
\(A + B + C + D + E +2x+2y+2z+2w+2v =900\)

\(540 + 2x+2y+2z+2w+2v = 900\)

\(2x+2y+2z+2w+2v = 360\)

\(x+y+z+w+v=180\)

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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:


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

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.


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.

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Solution:

To start, you should recognize that the sum of the exterior angles of any polygon is 360.

Let’s solve the question for the case where the pentagon formed in the center is regular. When that is the case, each of the 5 exterior angles of the pentagon equals 360/5 = 72. Then you’ll see that the vertical angles of the triangles are all equal, so each of those angles is also 72. Thus, x, y, z, w, and v, respectively, each equals 180 - 2 x 72 = 36. Finally, this means the sum of v + x + y + z + w is 36 x 5 = 180.

Now, notice that, even if the pentagon is not regular, neither the sum of the interior angles nor the relations between the angles of the pentagon and the angles x, y, z, w and v change. Therefore, the answer for the general case is also 180.

Alternative Solution:

Basically, the problem is asking for the total measure of the 5 marked angles, and yes, we should recognize that the sum of exterior angles of any polygon is 360 degrees. The 5 triangles that have the 5 marked angles also have 10 unmarked angles, which form two sets of exterior angles to the pentagon in the center. Therefore, the total measure of these 10 angles is 2 x 360 = 720 degrees. Therefore, we can say the following (TM means “total measure”):

TM of the angles of 5 triangles = TM of the 5 marked angles + TM of the 10 unmarked angles

5 x 180 = TM of the 5 marked angles + 720

900 = TM of the 5 marked angles + 720

180 = TM of the 5 marked angles

Answer: C

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