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# R is a convex polygon. Does R have at least 8 sides? (1)

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R is a convex polygon. Does R have at least 8 sides? (1) [#permalink]  23 Sep 2006, 08:56
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R is a convex polygon. Does R have at least 8 sides?

(1) Exactly 3 of the interior angles of R are greater than 80 degrees.
(2) None of the interior angles of R are less than 60 degrees.
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R is a convex polygon. Does R have at least 8 sides?

(1) Exactly 3 of the interior angles of R are greater than 80 degrees.
(2) None of the interior angles of R are less than 60 degrees.

What is convex polygon
A convex polygon is a simple polygon whose interior is a convex set. The following properties of a simple polygon are all equivalent to convexity:

* Every internal angle is at most 180 degrees.
* Every line segment between two vertices of the polygon does not go
exterior to the polygon

To have minimun 8 sides a polygon should have sum of angles 180(n-2)
180(8-2) = 1080

(1). Exactly 3 of the interior angles of R are greater than 80 degrees.
Lets take the greatest value for these angles- 180 and smallest for other 5 -80.
180*3+80*5= 940 < 1080 ..SUFFI

(2) None of the interior angles of R are less than 60 degrees.
Angles can have any value ..NOT SUFFI

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