Question

How to solve q 92 on the sides and angles of a polygon.?

Answer 1

Hence the number of sides of the convex polygon is `15` and magnitude of the angle excluded is `150^@`

Explanation:

Let the actual number of sides of the polygon be n. So the sum of n interior angles will be `(2n-4)*90^@`
Now if `x^@` be angle excluded to give the sum of `(n-1)` angles as `2190^@` As the polygon is convex one `x<180^@`

Hence we can write

`(2n-4)xx90-x=2190`

`=>(180n-360)-x=2190`

`=>180n=(2190+360+x)`

`=>n=(2550+x)/180`

`=>n=(180*14+(30+x))/180`

`=>n=14+(30+x)/180`

As `n` is a positive integer.Here it should be greater than `14`. For `x<180^@` Considering minimum value of `(30+x)/180=1` we get `x=150^@`

Hence the number of sides of the convex polygon is `15` and magnitude of the angle excluded is `150^@`

Answer 2

`15` sides . Option C

Explanation:

A polygon can be divided into triangles by joining one vertex to all the others, The number of triangles will be:

`"sum of interior angles"/(180°)`

For example: `(1080°)/(180°) = 6` triangles `hArr 8` sides

In this case: `(2190°)/(180°) = 12.167` triangles which is obviously not possible as the answer has to be a whole number.

The polygon is convex which means that the missing angle is not a reflex angle.

Therefore there must be `13` triangles which means `15` sides.

To find the size of the missing angle:

The sum of the interior angles will be:

`13 xx180°= 2340°`

The missing angle is `2340° -2190° = 150°`