measure of one external angle of a regular polygon is grater than the measure of one internal angle
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8th
Maths
Understanding Quadrilaterals
Angle Sum Property
Let the formula relation th...
MATHS
Let the formula relation the exterior angle and number of sides of a polygon be given as nA=360.
The measure A, in degrees, of an exterior angle of a regular polygon is related to the number of sides, n, of the polygon by the formula above. If the measure of an exterior angle of a regular polygon is greater than 50, what is the greatest number of sides it can have?
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ANSWER
Sum of exterior angles for any polynomial is always 360.
Since polynomial has n angles, each with exterior angle is A, then
sum of exterior angles will be nA
Given, nA=360
∴A=
n
360
We are given that: A>50
⇒
n
360
>50
⇒360>50n
⇒n<
50
360
⇒n<7.2
Hence, the greatest number of angles polygon can have is 7.
A triangle. The measure of each exterior angle in a regular polygon is 360°/n, where n is the number of sides.
The measure of the internal angle and its adjacent external angle is always 180°.
In a regular triangle the exterior angles are each 360°/3, or 120°. The interior angles are 180° - 120°, or 60°. This makes the exterior angle greater than the interior angles.
In a regular quadrilateral(square) the exterior angle is 360°/4 or 90°. The interior angles are each 180° - 90°, or 90°. So neither internal nor external angle is bigger than the other.
In any other regular polygon the exterior angles will be less than 90°, so the internal angles will be greater than 90°. So only a regular triangle has larger exterior angles than internal angles.