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Geometry: Decagon

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by Yaj » Sat Sep 14, 2013 6:02 am
Some help!!

The figure shows a regular decagon (10-side polygon). Angle x is formed by extending two of the sides of the decagon. What is the degree measure of angle x?
(A) 96
(B) 108
(C) 120
(D) 144
(E) 150

[spoiler]OA:B[/spoiler]

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by vinay1983 » Sat Sep 14, 2013 6:28 am
The interior angles of the decagon is equal to (n-2)*180, where n is equal to no.of sides

So, (10-2)*180 = 8 * 180 = 1440 i.e 1440/10 = 144 degrees. So let us name the triangle such formed(as per the figure) A B C in clockwise direction.

As per above each angle is equal to 144 degrees, so we know that on a straight line the angle is equal to 180 degrees so, 144+x=180 i.e x = 36 degrees (interior angle CAB and likewise angle BCA).

We also know that sum of interior angles of a triangle is 180 degrees, hence in triangle ABC angle (x as per your diagram)ABC + angle BAC + angle BCA = 180 degrees

Of this we know the value of angle CAB and angle BCA values, hence angle ABC = 180 - 2*36 = 180-72=108 degrees

Option B is correct.


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Geometry: Decagon

by Brent@GMATPrepNow » Sat Sep 14, 2013 6:31 am
Image
The figure shows a regular decagon (10-side polygon). Angle x is formed by extending two of the sides of the decagon. What is the degree measure of angle x?
(A) 96
(B) 108
(C) 120
(D) 144
(E) 150
Nice rule: The sum of the interior angles in an N-sided polygon is equal to 180(N - 2) degrees

So, in a 10-sided polygon, the sum of the interior angles = 180(10 - 2) = 1440 degrees

Since the polygon is a REGULAR polygon, all 10 sides have the same length, and all 10 angles are equal.
So, if the sum of all 10 angles = 1440 degrees, then each angle must = 1440/10 = 144

So, we get:
Image

At this point, we can use the fact that angles on a line add to 180 degrees to conclude that the two angles inside the triangle are 36 degrees each.
Image

So, 36 + 36 + x = 180
x = 108

Answer: B

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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