Question

The angle of a triangle are in A.P. if the greatest angle is twice the least,find all angles

Answer 1

Let the angles of the triangle which are in A.P be ;
a,b, & c
_________________
b = a+d
c = b+d

2b = a+c
b = (a+c)/2 -------------(1)
___________________
According to the question ;
c = 2a -------------------(2)

b = (a+2a)/2
b = (3/2)a----------------(3)

_____________________
And :
a+b+c = 180° ------(4)
------(angle sum property of triangle)
========================
a+(3/2)a+2a = 180
3a + (3/2)a = 180
a + (1/2)a = 60
(3/2)a = 60
a = 20×2
-------------------------------------
a = 40°
-------------------------------------
b = (3/2)a = 60°
-----------------–------------------
c = 2a = 80°
____________________

Answer 2

let a,b,c, are in ap
so 2b = a+c
also given greatest angle = twice the least
                          c = 2a
so 2b = a+2a
    2b = 3a
      b =3a/2
we know that sum of angles in triangle = 180
                   a+b+c = 180
                   a+3a/2+2a = 180
                     9a/2 = 180
                        9a = 360
                          a = 40
so b = 3a/2
        = 3(40)/2
    b = 60
so c = 2a 
        = 2(40)
     c = 80
the angles of triangle are 40,60,80 are in AP with common difference 20