Question

the angles of a quadrilateral are in A.P. least angle is one-fourth the greatest angle. find the least angle

Answer 1

Let the 4 angles be a,a+d,a+2d and 4a (Largest angle is 4 times smallest)

Also, with common difference d,

4a=a+3d

Thus, d=a

So AP is a,2a,3a,4a

Sum of angles in quadrilateral should be 360

a+2a+3a+4a=360

10a=360

a=36

Least angle is 36 deg

Answer 2

Given : angles of a quadrilateral are in ap the least angle is one fourth of the greatest angle.

To  find  : least angle

Solution:

Let say 4 angles in AP  are

a , a + d  , a + 2d    , a + 3d

Sum of angles of a quadrilateral is 360°

=> a +  a + d + a + 2d + a + 3d  = 4a + 6d  = 360

=> 2a  + 3d  =  180

 least angle is one fourth of the greatest angle

=> a  = (1/4)(a + 3d)

=> 4a  = a + 3d

=> 3a  = 3d

=> a   = d

2a  + 3d  =  180

=> 2a  + 3a  = 180

=> 5a  = 180

=> a   = 36

a   = d => d = 36

Hence angles of the quadrilateral​

36°  , 72° , 108°  , 144°

least angle  36°

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