Let other two angles of the triangle be a and b and a+z=180 and y+b=180
w=a+b
z=b+x
y=a+x
From 1
w+z=240
or a+b+b+x=240
or 180+b=240
or b=60..no information about x...not sufficient

From 2
a+b=b+x or a=x
b+x=a+x or b=a
so a=b=x=60..sufficient

Ans option B

Let the supplemmentary angles of W, Y and Z = x, y and z these are the internal angles of the triangle.
2) W=Y=Z
W= z+y , Y = x+z and Z = x+y
W+Y+Z = 2(x+y+z)
3W = 2*180°
W = 2*60 = 120
so x = 180 - 120 = 60 Suff.
1) W+Z = (y+z)+(x+y) = (x+y+z) + y = 180+y = 240
y = 60
Insuff
IMO B

I think the answer is B.

From the figure, we can say,
w = 180 - x ==> x = 180 - w.
x + 180 - y + 180 - z = 180 further solving we will get 180 + x - y - z = 0
substitute x = 180 - w, then, we will get
180 + 180 - w - y - z = 0
w + y + z = 360

Now look at the two statements sufficient or not.

Statement - 1
Given, w + z = 240, now substitute the same in w+y+z=360

then, we will get y=120, so that we can rewrite the sentence as,
x+ 180 - y + 180 - z = 180
substitute y = 120, then
x + 60 + 180 - z = 180
x - z = - 60 which we end up for calculating x , we need a value of either w or x.
So, statement-1 is NOT SUFFICIENT.

Statement - 2
Given w=y=z, then
As we have condition w+x = 180, we should get the equation by substituting w in place of y & z in the second equation, by that we can find x value, lets see,
the second equation is x + 180 - y + 180 - z = 180
then, x - y - z = -180, substitute w in place of y & z.
x - w - w = -180
x - 2w = -180 ( substitute w = 180 - x )
x - 2(180 - x) = -180
x - 360 +2x = -180
3x = 180
x = 60. It is UNIQUE, hence the statement is SUFFICIENT.

Then Answer is B (statement-2 alone is SUFFICIENT).

HTH, GOOD LUCK,

Thanks,
Rajesh,
Loves GMAT....!!!!