 QUESTION

# Which of the following statements is not correct?A.If the exterior angle of a regular polygon is $30{}^\circ$ , it has 12 sides.B.If the interior and exterior angle of a regular polygon are equal, it is a rectangle.C.If the exterior angle of a regular polygon is greater than its interior angle, then it is an equilateral triangle.D.In a regular pentagon, the exterior angle is half of the interior angle.

Hint:First, try to find a general relation between the number of sides and the exterior angle of the regular polygon.Let us consider a regular polygon of side n. Also, let the interior angle to be i and the exterior angle to be y.Therefore, the number of interior angles in an n-sided polygon is n.

Now, we know that the sum of the interior and the exterior angle of a polygon is $180{}^\circ$ . Representing this mathematically, we get
Interior angle + exterior angle = $180{}^\circ$
$i+e=180{}^\circ .........(i)$
The other thing we know is, for a regular polygon, all the sides are equal, and the measure of all interior angles are also equal. We also know that the sum of all the interior angles of an n-sided polygon is given by $\left( n-2 \right)180{}^\circ$ . So, representing all the above data in form mathematical equation, we get
$ni=\left( n-2 \right)180{}^\circ$
$\Rightarrow i=\dfrac{\left( n-2 \right)180{}^\circ }{n}........(ii)$
Now we will substitute the value of i in equation (i).
$i+e=180{}^\circ$
$\Rightarrow \dfrac{\left( n-2 \right)180{}^\circ }{n}+e=180{}^\circ$
$\Rightarrow e=180{}^\circ -\dfrac{\left( n-2 \right)180{}^\circ }{n}........(iii)$
Now we will start with the checking of the options. First, lets us start with option (a). On putting n = 12 in equation (iii) the exterior angle comes out to be:
$e=180{}^\circ -\dfrac{\left( 12-2 \right)180{}^\circ }{12}=180{}^\circ -150{}^\circ =30{}^\circ$
So, as the exterior angle is $30{}^\circ$ option (a) is correct.
Now, let us check option (b). We know that the interior angles of a rectangle are $90{}^\circ$ .So, putting i in the equation, we get
$90{}^\circ +e=180{}^\circ$
$\Rightarrow e=90{}^\circ$
Therefore for a rectangle e and i are equal, making statement (b) correct.
Now, let us check option (c). According to the statement:
e>i
Now we will put the values of e and I from equations (ii) and (iii).
$180{}^\circ -\dfrac{\left( n-2 \right)180{}^\circ }{n}>\dfrac{\left( n-2 \right)}{n}180{}^\circ$
$\Rightarrow 1-\dfrac{\left( n-2 \right)}{n}>\dfrac{n-2}{n}$
$\Rightarrow n-n+2>\dfrac{n\left( n-2 \right)}{n}$
$\Rightarrow 2>n-2$
$\Rightarrow n<4$
Therefore, the only possible value of n is 3 so option (c) is correct.
Finally, checking for option (d). According to the statement for a pentagon: $e=\dfrac{1}{2}i$
So, putting n=5 in equations (ii) and (iii), we get
$i=\dfrac{\left( 5-2 \right)180{}^\circ }{5}=108{}^\circ$
$e=180{}^\circ -\dfrac{\left( 5-2 \right)180{}^\circ }{5}=180{}^\circ -108{}^\circ =72{}^\circ$
But this violates the statement given in the option (d).
Therefore, the wrong statement among the four statements is of option (d).

Note: Also, remember that the sum of all the exterior angles of a polygon is always equal to $360{}^\circ$ , no matter whether it is a regular polygon or not.