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06 Oct 2020, 04:19
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Bunuel
PLUG IN for the angles.
In the center of the star is a PENTAGON.
For any polygon with n sides, the sum of the interior angles \(= (n-2)(180)\)
Thus, the sum of the angles inside the pentagon \(= (5-2)(180) = 540\)
Let each of the 5 angles inside the pentagon \(= \frac{540}{5} = 108\)
Angles that form a straight line and angles inside a triangle must both sum to 180.
As a result, the following figure is yielded:
Attachment:
star.jpg [ 7.04 KiB | Viewed 15084 times ]
18 Apr 2021, 15:48
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Hi All,
To start, it’s incredibly rare to see a Quant question on the GMAT with so many inter-connected shapes (including a hexagon!), so you shouldn’t worry about this question if you got it wrong the first time you attempted it. You can solve it with a mix of Geometry rules and TESTing VALUES.
With geometric shapes, every time you “add a side”, you increase the total number of degrees by 180. For example….
A triangle = 180 degrees
A square/rectangle = 360 degrees
A hexagon (5-sided shape) = 540 degrees
Etc.
Since we are not given any information about any of the angles in this picture, we can TEST VALUES. Since the hexagon includes 5 angles that total 540 degrees, it’s easiest to make all 5 angles the same…. 540/5 = 108 degrees each.
The sum of the angles on a line add up to 180 degrees, so each angle in each of the triangles that is next to a 108 degree hexagon angle is equal to 72.
That means that each triangle is an isosceles triangle with two 72 degree angles and one angle that equals… 180 – 72 – 72 = 36 degrees. Thus, V = W = X = Y = Z = 36 and the sum of those 5 angles is (5)(36) = 180 degrees.
Final Answer:
GMAT assassins aren't born, they're made,
Rich
To start, it’s incredibly rare to see a Quant question on the GMAT with so many inter-connected shapes (including a hexagon!), so you shouldn’t worry about this question if you got it wrong the first time you attempted it. You can solve it with a mix of Geometry rules and TESTing VALUES.
With geometric shapes, every time you “add a side”, you increase the total number of degrees by 180. For example….
A triangle = 180 degrees
A square/rectangle = 360 degrees
A hexagon (5-sided shape) = 540 degrees
Etc.
Since we are not given any information about any of the angles in this picture, we can TEST VALUES. Since the hexagon includes 5 angles that total 540 degrees, it’s easiest to make all 5 angles the same…. 540/5 = 108 degrees each.
The sum of the angles on a line add up to 180 degrees, so each angle in each of the triangles that is next to a 108 degree hexagon angle is equal to 72.
That means that each triangle is an isosceles triangle with two 72 degree angles and one angle that equals… 180 – 72 – 72 = 36 degrees. Thus, V = W = X = Y = Z = 36 and the sum of those 5 angles is (5)(36) = 180 degrees.
Final Answer:
GMAT assassins aren't born, they're made,
Rich
08 Jul 2021, 01:15
Kudos
Bookmarks
You have 5 triangles extending from each side of the pentagon.
The total degree measure of all 5 triangles is = (5) * (180 deg. per triangle) = 900 degrees
Rule: the SUM of all the exterior Angles of ANY polygon (taken ONCE for each vertex) will always be = 360 deg.
In the figure, we are taking the exterior angels TWICE at each vertex. These are the other 10 interior angles inside the five triangles that are *NOT* marked with a variable.
thus, the SUM of every other Interior angle inside the 5 triangles EXCEPT v + w + x + y + z
=is equal to=
the SUM of the exterior angles of a polygon taken TWICE = (2) (360) = 720 deg.
The remaining angles will simply by:
(900 deg. measure of all 5 triangles)
-
(720 deg.)
__________
180
Posted from my mobile device
The total degree measure of all 5 triangles is = (5) * (180 deg. per triangle) = 900 degrees
Rule: the SUM of all the exterior Angles of ANY polygon (taken ONCE for each vertex) will always be = 360 deg.
In the figure, we are taking the exterior angels TWICE at each vertex. These are the other 10 interior angles inside the five triangles that are *NOT* marked with a variable.
thus, the SUM of every other Interior angle inside the 5 triangles EXCEPT v + w + x + y + z
=is equal to=
the SUM of the exterior angles of a polygon taken TWICE = (2) (360) = 720 deg.
The remaining angles will simply by:
(900 deg. measure of all 5 triangles)
-
(720 deg.)
__________
180
Posted from my mobile device
dylanl1218
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20 Jul 2021, 08:05
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Bunuel
Definitely a challenging question.
Notice that each angle in the pentagon corresponds to a triangle with two of the stated variables from the stem. So each angle of the polygon can be calculated as 180 - Variable from Angle 1 - Variable from Angle 2.
Now remember that 180(N-2) in which N = # of sides of a polygon is equal to the SUM of the angles in a polygon. So 180(N-2) = 540. Finally we can sum up each of the calculated polygon angles and set THAT equal to 540. So 900-2(v+x+y+z+w) = 540. From here we can calculate that the sum of the variables is equal to 180.
Attachments
ToughGraphExample.png [ 19.5 KiB | Viewed 8016 times ]
