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Analytical Geometry

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6 Dec 2009 22:36:23 IST

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1. The sum of the interior angles of a polygon is three times the sum of its exterior angles. Th

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1. The sum of the interior angles of a polygon is three times the sum of its exterior angles. Then number of sides in a polygon is(a) 6(b) 7(c) 8(d) 9

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Kshitij Karandikar

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10 Dec 2009 10:51:26 IST

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The Answr should be 4. Its one exterior angle is of 270, so sum of all exterior angles is 1080. It is thrice to 360 , the sum of all interior angles...

edison

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10 Dec 2009 11:53:59 IST

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Sum of Angles in Polygons

Introduction: A polygon is a closed plane (flat) figure with straight sides. There is a way to measure the sum of the interior angles that depends upon the number of sides (which is the same as the number of angles). We show this and also investigate the exterior angles.

Since the sum of the angles in a triangle is 180º, the sum of the angles in the quadrilateral is 360º because it is composed of two triangles. Similarly, we see that the sum of the five angles in the pentagon is 540º since it is composed of three triangles and 3 x 180º = 540º.

The number of triangles which compose the polygon is two less than the number of sides (angles). We generalize this result for a polygon of n sides and angles:

Theorem: The sum of the interior angles in a polygon with n sides is 180º(n – 2).

In the pentagon below, we have labeled the interior angles 1, 2, 3, 4, and 5. Each of these is supplementary respectively to exterior angles 6, 7, 8, 9, and 10. Therefore we have:

We know that angles 1 through 5 in a pentagon have a sum of 540º. We substitute 540º for these angles and we have:. Subtracting 540º from both sides, we can find the sum of the five exterior angles of this pentagon:
.
The sum of these exterior angles in any polygon will always be 360º, and although this is not a complete proof, we state the following:

Theorem: The sum of the exterior angles of a polygon is 360º.

edison

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10 Dec 2009 11:55:35 IST

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Now using the above theorems that

1)Theorem: The sum of the interior angles in a polygon with n sides is 180º(n – 2).

2) Theorem: The sum of the exterior angles of a polygon is 360º.

Thus 180º(n – 2) / 360º = 3

or n-2 = 3*2

or n = 8

Thus correct option is (c) 8

Deevita Agarwal

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1 Mar 2010 22:05:45 IST

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Thanks

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