Question

The angles of a pentagon is in arithmetic sequence.. prove that uts smallest angle is greater than 60 degree

Answer 1

The proof is given below:

Note: In the question, it should be hexagon.

Let the smallest angle be a

This will be the first term of the AP

If the common difference is d then

The angles will be

$a,a+d,a+2d,a+3d,a+4d,a+5d$

A hexagon can be divided into 4 triangles

thus, the sum of all the interior angles of the pentagon will be

$4\times 180^\circ$

$=720^\circ$

or $a+(a+d)+(a+2d)+(a+3d)+(a+4d)+(a+5d)=720^\circ$

$\implies 6a+15d=720^\circ$

$\implies 2a+5d=240^\circ$

if we take $a=60^\circ$

Then

$2\times 60^\circ+5d=240^\circ$

$\implies 5d=120^\circ$

$\implies d=24^\circ$

Therefore, the angles will be

$60^\circ,84^\circ,108^\circ,132^\circ,156^\circ,180^\circ$

Thus, in this case one of the angles is $180^\circ$

But in a polygon no interior angle can be $180^\circ$

Therefore, the first angle must be greater than $60^\circ$

Hope this answer is helpful.

Know More:

Q: The angles in a nine sided polygon are in the arithmetic sequence. Is 100 degree the smallest angle of the polygon? Justify ur answer

Click Here: https://brainly.in/question/9009987

Q: The angle measures of an octagon are in arithmetic sequence.

(a) What is the sum of its angles ?

(b) What is the sum of its smallest and largest angles ?

(c) If the difference between the smallest and largest angles is 70°, what is the measure  of smallest angle?​

Click Here: https://brainly.in/question/12291411

Answer 2

Given :  smallest angle is greater than 60 degree in hexagon

To find : prove that its smallest angle is greater than 60 degree

Solution:

Correction :

Question should have hexagon

Sum of angle of polygon of n sided = (n- 2) * 180°

Hexagon has 6 sides  so

Sum of all angles = (6 - 2) * 180° = 720°

Let say smallest angle =  a°      a  > 0

and d° is the common difference  

then largest angle = a + 5d

largest angle should be less than 180°

=> a + 5d <  180°

=> a + 5d = 180 - k   k > 0

Sum of all angles

a + a + d  + a + 2d + a + 3d + a + 4d + a + 5d  = 720

=> 6a + 15d  = 720

=> 2a + 5d  = 240

=> a  + a + 5d  =  240

=> a  + 180 - k  =  240

=> a = 60 + k

=> a > 60°

QED

Hence proved

smallest angle is greater than 60°

for

5a + 10d = 540

=> a + 2d = 108

=> 2a + 4d = 216

=> a + a + 4d = 216

a + 4d <  180

=> a  > 36 for pentagon

Example : 48 , 78 , 108 , 138 , 168    is one of  the possible angles where

smallest angle is less than 60 degree.

Learn more:

find the number of sides of a regular polygon when each of its ...

brainly.in/question/7776593

What is the angle between two adjacent sides of a regular polygon ...

brainly.in/question/3232652