CLASS-8
ANGLE PROPERTIES OF A POLYGON

ANGLE PROPERTIES OF A POLYGON -

Properties.1) The sum of the Interior angles of a polygon with ‘n’ sides is considered as = (2n – 4) Right Angles.

For your better understanding, there are some examples are given below –

Example.1- The sum of the interior angles of a triangle = (2 X 3 – 4) right angles = 2 right angles = 180⁰, which is known to be true

Example.2- The sum of the interior angles of a quadrilateral = (2n – 4) right angle 

= (2 X 4 – 4) right-angle = 4 right angles

In the adjoining figure, the sum of the angles of the quadrilateral ABCD = the sum of the angles of ∆ BAC + the sum of the angles of ∆ ADC = 2 right angles + 2 right angles = 4 right angles.

So, it can be concluded that the property is true for a quadrilateral

Example.3- The sum of the interior angles of a Hexagon =  (2n – 4) right angle

                                  = (2 X 6 – 4) right-angle = 8 right angles

In the adjoining figure, the sum of the angles of the hexagon ABCDEF = the sum of the angles of a triangles AFE, AED, ADC, and ACB = 4 X 2 right angles = 8 right angles

So, the property holds for a hexagon and it can be verified easily that it holds for other polygons as well.




Properties.2)  The sum of the exterior angles of a convex polygon = 360⁰

Example.1) In the adjoining figure,

∠a + ∠1 = 180⁰, ∠b + ∠2 = 180⁰, ∠c + ∠3 = 180⁰

∠a + ∠1 + ∠b + ∠2 + ∠c + ∠3 = 180⁰ + 180⁰ + 180⁰ = 540⁰

=> ∠a + ∠b + ∠c + ∠1 + ∠2 + ∠3 = 540⁰

=> ∠a + ∠b + ∠c + (∠1 + ∠2 + ∠3) =  540⁰

But,  ∠1 + ∠2 + ∠3 = 180⁰, being angles of a triangle.

So, ∠a + ∠b + ∠c  =  540⁰ - 180⁰ =  360⁰

So, the property is true for a triangle.



Example.2)  In the adjoining figure,

∠a + ∠1 = 180⁰, ∠b + ∠2 = 180⁰, ∠c + ∠3 = 180⁰, ∠d + ∠4 = 180⁰

So, ∠a + ∠1 + ∠b + ∠2 + ∠c + ∠3 + ∠d + ∠4 = 180⁰ + 180⁰ + 180⁰ + 180⁰

So, ∠a + ∠b + ∠c + ∠d + (∠1 + ∠2 + ∠3 + ∠4) = 720⁰

But, the sum of the interior angles of a quadrilateral =  360⁰

So,  ∠a + ∠b + ∠c + ∠d + 360⁰ =  720⁰

So,  ∠a + ∠b + ∠c + ∠d  =  720⁰ - 360⁰ = 360⁰

So, the property holds for a quadrilateral.



Example.3)  In the adjoining figure,

∠a + ∠1 = 180⁰, ∠b + ∠2 = 180⁰, ∠c + ∠3 = 180⁰, ∠d + ∠4 = 180⁰, ∠e + ∠5 = 180⁰

So, ∠a + ∠1 + ∠b + ∠2 + ∠c + ∠3 + ∠d + ∠4 + ∠e + ∠5 =  180⁰ + 180⁰ + 180⁰ + 180⁰ + 180⁰


So, ∠a + ∠b + ∠c + ∠d + ∠e + ∠1 + ∠2 + ∠3 + ∠4 + ∠5  =  900⁰

So,  ∠a + ∠b +  ∠c + ∠d + ∠e + (∠1 + ∠2 + ∠3 + ∠4 + ∠5)  =  900⁰

But, the sum of interior angles of a pentagon

                          =  (2 X 5 – 4) right angles =  6 X 90⁰ = 540⁰

So,  ∠a + ∠b +  ∠c + ∠d + ∠e + (∠1 + ∠2 + ∠3 + ∠4 + ∠5) =  900⁰

So,  ∠a + ∠b +  ∠c + ∠d + ∠e + 540⁰ =  900⁰

So,  ∠a + ∠b +  ∠c + ∠d + ∠e =  900⁰ - 540⁰ =  360⁰

So, the property is true for the pentagon.

We can easily verify that the property holds for other polygons as well.