the difference between the measure of interior and exterior angles of regular polygon is 90 degrees?

Question

sir n=?? and explain me some examples to find n?i cant able to do sir plzzzzzzzzzzzz help me

Answer 1

This is a little complicated.

(n-2)180 gives the sum of the interior angles of a regular polygon

[(n-2)180]/n gives the measure of each interior angle

180-[(n-2)180]/n gives the measure of the corresponding exterior angle, which becomes

(180n-180n+360)/n=(360/n)

(360/n) gives the exterior angle

you want this difference to be 90º

[(n-2)180]/n-(360/n)=90

multiply both sides by n

(n-2)180-360=90n

180n-360-360=90n

180n-720=90n

180n-90n-720=0

180n-90n=720

90n=720

n=720/90

n=8

the figure is an octagon

(n-2)180=(8-2)180=6*180=1080º for the sum of the interior angles

1080/8=135, each interior angle of an octagon is 135º

therefore each corresponding exterior angle is 180-135=45º

135º-45º=90º which is what was asked for

If you find this hard to follow, use trial and error.

(n-2)180

let n=6, (6-2)180=720, 720/6=120, 180-120=60, 120-60=60, not 90

let n=7, (7-2)180=900, 900/7=128 4/7, 180-128 4/7=51 3/7, with the difference being 77 1/7, not 90

let n=8, (8-2)180=1080, 1080/8=135, 180-135=45, 135-45=90 which is what you asked for.

 

Answer 2

The sum of the exterior angles of all polygons is 360°.

The sum of the interior angles of a polygon is (n - 2)180°. n is the number of sides.

The triangle is the polygon with the least number of sides. The interior angles add to 180°

The smallest difference between the exterior and interior angles would be 180°

90° is not possible.

 

Check your numbers.