Geometry: Decagon

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Yaj: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Tue Jul 09, 2013 12:53 pm; Thanked: 1 times

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]

Thank You!

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Source: Beat The GMAT — Problem Solving | Sat Sep 14, 2013 6:02 am

vinay1983: Legendary Member; Posts: 643; Joined: Wed Aug 14, 2013 4:27 amFollowed by: 7 members; Thanked: 48 times

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.

I hope I am right!

You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

Geometry: Decagon

by Brent@GMATPrepNow » Sat Sep 14, 2013 6:31 am

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:

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.

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

Answer: B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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