Question

difference between any two consecutive interior angles of a polygon is 5 degree if the smallest angle is 120 degree find the number of sides of a polygon

Answer 1

Answer:

16 or 9

Step-by-step explanation:

Let number of sides = n

∴ number of interior angles = n

Given that difference between two consecutive angles is 5 and smallest angle is 120°, we get angles in following manner

120, 125, 130..... upto n

Here you can observe that angles are forming an AP where

First Term (a)                  = 120

Common Difference (d) = 5

Number of terms           = n

We know that sum of all interior angles = 180 × ( n - 2) ----- ( i )

Also, Sum of all angles in AP

 $=\frac{n}{2}[2a+(n-1)d]--------(ii)$

From ( i ) and ( ii )

$180\times(n-2)=\frac{n}{2}[2a+(n-1)d]$

Putting values of a = 120, d = 5

$180\times(n-2)=\frac{n}{2}[240+(n-1)5]$

360 ( n - 2) = n [ 240 + 5n - 5]

360n - 720 = n [ 235 + 5n]

360n - 720 = 235n + 5n²

5n² -125n + 720 = 0

Dividing whole equation by 5

n² - 25n + 144 = 0

Splitting the middle term

n² - 16n - 9n + 144 = 0

n ( n - 16) -9( n - 16) = 0

(n - 16) ( n - 9) = 0

n - 16 = 0          n - 9 = 0

n = 16               n = 9

∴ Number of sides in the given polygon can be 16 or 9.

Answer 2

Step-by-step explanation:

hope it helps you ✌✌✌