ajit257 wrote:

Karishma,

Can you explain in more details when you say "The average of the given angles can only match 140 (such that effectively, all the angles are 140)". I did not get your point. Thanks

Sum of interior angles of a polygon = (n-2)*180 (not necessarily regular polygon)

Why? See the figure below:

Attachment:

Ques2.jpg [ 5.14 KiB | Viewed 4638 times ]
A 6 sided polygon can be split into 4 triangles each of which has a sum of interior angles 180 degrees.

An n sided polygon can be split into n - 2 triangles.

When the polygon is regular, each angle is same so the sum is divided by the number of sides to get the measure of each angle e.g. in a regular hexagon, each interior angle = 4*180/6 = 120 degrees.

Now if I have a hexagon whose angles are 115, 117, 119, 121, 123 and x, what will be the angle x?

We can see it in two ways -

1. The sum of all angles should be 4*180 = 720

So 115 + 117 + 119 + 121 + 123 + x = 720

or x = 125

2. The average of the angles should be 120. (Since the sum of the angles is 720 and there are 6 sides)

119 and 121 average out as 120. (119 is 1 less than 120 and 121 is 1 more than 120)

117 and 123 average out as 120.

So 115 and x should average out as 120 too. Therefore, x should be 125.

In the question, the average of the given angles of the polygon can only be 140. So it must have 9 sides. To confirm,

136, 137, 138, 139, 140, 141, 142, 143, 144 - 9 angles with average 140. So the polygon must have 9 sides.

(It cannot be 144 or anything else because 10 angles (136, 137, 138, 139, 140, 141, 142, 143, 144, 145) will not give an average of 144)

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Karishma

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