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




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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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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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As shown in the diagrams above, nothing about the colored angles can be assumed. So I considered triangles that forms the interior angles of the inside pentagon. Sum of those individual angles = 360

Bases on the above method I am getting 270 as my answer

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

could you please elaborate on this method. very interesting approach.

the highlighted part.... have a difficult time to apply the theory properly.

Very nice solution man..!

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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