earth@work wrote:The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120° , find the number of the sides of the polygon.
a)5 b)13 c)33 d)7 e)9

The sum of the angles in an n-sided shape is (n-2)*180. We know:

120 + 125 + 130 + ... = (n-2)*180

If we have n angles, the largest angle will be 120 + (n-1)*5 = 115 + 5n. We're adding an evenly spaced series here, so we can find the sum, using the average fomula:

sum = (average)*(number)

Because the series is evenly spaced, the average of the series is equal to the average of the smallest (120) and the largest (115 + 5n). There are n terms, so:

sum = [(120 + 115 + 5n)/2]*n
(n-2)*180 = (235 + 5n)*n/2
360n - 720 = 235n + 5n^2
72n - 144 = 47n + n^2
0 = n^2 - 25n + 144
0 = (n-16)(n-9)
n = 9 or n = 16.

We must discard the n=16 solution, because if n=16, the angles go past 180, and you can't have a 180 degree angle as an interior angle in a polygon- it makes a straight line.

earth@work wrote:Hi Ian,
Thanks, but there is one confusion i still have, cud u pls make me understand -i think i m making some silly mistake..

We are taking 120, 125,130, 135 ..... as the consecutive angles which forms A.P.
the angle consecutive to 120 angle i.e angle on both the sides of 120 angle shud be 125 degrees ....this gives us two 125 degree angle...
similarly for both 125 degree angle-if there is 120 on one side then shud be 130 degree angle on the other side of both the angles..i.e we get two 130degree angle?
what m i missing, pls help

Great point- the question actually doesn't make sense, at least as far as I can tell. In my solution (which seems to be the one intended by the question designer), the angles work out to be, going around the polygon, 120, 125, 130, 135, 140, 145, 150, 155, 160. Of course, then you have a 160 degree angle next to a 120 degree angle, which violates the conditions in the question. Still, if you try to make it so that there is a 5 degree difference all around the polygon (something like 120, 125, 130, 135, 140, 135, 130, 125 would work, except that it doesn't add up to the right sum), then there is no integer solution for the number of sides, unless I made an error with the algebra (it gets a bit complicated, so that's a possibility). So I think the question is not well designed (or I've made some kind of error), and the wording should be changed. I'm pretty sure the intended question is something more like this:

In a polygon, the smallest angle, A, is adjacent to the largest angle, B. With the exception of the difference between A and B, the difference between any two consecutive interior angles of this polygon is 5°. If the measure of angle A is 120 degrees, find the number of the sides of the polygon.

Then my solution above is correct. But as the original question was worded, there doesn't seem to be any solution. Where is the question from?