Induction (1 Viewer)

[Lukybear]

THe sum of the angles of a polygon of n sides is (2n-4) right angles, n>=3

 

[Trebla]

Let S_n = (2n - 4)90º be the angle sum of a n-sided polygon
When n = 3
LHS = S₃ = 180º
Since we know that the angle sum of a three sided polygon (triangle) is always 180º
RHS = (2) x 90º = 180º
Hence statement holds for n = 3
Assume the truth of n = k
i.e. S_k = (2k - 4)90º
Required to prove case for n = k + 1
i.e. S_k+1 = (2k - 2)90º
LHS = S_k+1
= S_k + 180º
(Note that to form an extra side, we can place a triangle with its base on one existing side (draw a diagram to see this) which means an additional 180º is provided in the angle sum)
= (2k - 4)90º + 180º by assumption
= (2k - 4)90º + 2(90º)
= (2k - 2)90º
= RHS
Statement holds for n = k + 1 if it holds for n = k
Since it is holds for n = 3 then by induction it holds for all integers n > 3