Theorem 1

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In a polygon of 'n' sides, the sum of the interior angles is equal to (2n - 4) right angles.

ABCDE is an n sided polygon.

The sum of the interior angles = (2n - 4) right angles

Take any point O inside the polygon. Join OA, OB, OC.

Sum of 'n' interior angles = (2n - 4) 90^o.

Each interior angle of a regular polygon

[Since in a regular polygon, all the interior angles are equal].

If the sides of a convex polygon are produced in order (clockwise or anticlockwise), the sum of the exterior angles is 4 right angles.

= 4 right angles = 4 90^o

Each exterior angle for a regular polygon of n sides

Number of sides of a regular polygon

2. At each vertex, one interior angle + one exterior angle = 180^o.

How many sides has a regular polygon with each interior angle equal to 144^o?

[Each interior angle + each exterior angle = 180^o]

144^o + each exterior angle = 180^o

each exterior angle = 180^o - 144^o

= 36^o.

Number of sides

= 10

The sum of the interior of a regular polygon is 1080^o. How many sides are there?

The sum of the interior angles

= (2n - 4) x 90^o

= 1080^o

(2n - 4) x 90 = 1080

2n - 4 =

2n = 16

n = 8

The polygon has 8 sides.