# ${\text{(a)}}$ Is it possible to have a regular polygon with measure of each exterior angle as ${\text{2}}{{\text{2}}^0}$?

${\text{(b)}}$ Can it be an interior angle of a regular polygon? Why?

Answer

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Hint- Here, the property of a regular polygon is used (i.e., the sum of all the exterior angles which are equal to each other of a regular polygon is always ${360^0}$).

${\text{(a)}}$ No, it is not possible to have a regular polygon with measure of each exterior angle of ${\text{2}}{{\text{2}}^0}$ because ${\text{2}}{{\text{2}}^0}$ is not a multiple of ${360^0}$. Since, the sum of all the exterior angles of a regular polygon is always ${360^0}$ and all the exterior angles of a regular polygon are equal in measure.

${\text{(b)}}$ If the interior angle of a regular polygon is ${\text{2}}{{\text{2}}^0}$, then the measure of exterior angle of that regular polygon will be \[\left( {{{180}^0} - {\text{2}}{{\text{2}}^0}} \right) = {158^0}\]. Clearly, \[{158^0}\] is not a multiple of ${360^0}$. So, it is not possible to have a regular polygon with a measure of each interior angle of ${\text{2}}{{\text{2}}^0}$.

Note- In these types of problems, if the interior angle of the regular polygon is given then it is converted into the exterior angle of the regular polygon. Then using properties of a regular polygon like the sum of all the exterior angles is always equal to ${360^0}$ and each exterior angle is equal, we have to check whether these properties hold true or false. If they hold then that regular polygon is possible else, it is not.

${\text{(a)}}$ No, it is not possible to have a regular polygon with measure of each exterior angle of ${\text{2}}{{\text{2}}^0}$ because ${\text{2}}{{\text{2}}^0}$ is not a multiple of ${360^0}$. Since, the sum of all the exterior angles of a regular polygon is always ${360^0}$ and all the exterior angles of a regular polygon are equal in measure.

${\text{(b)}}$ If the interior angle of a regular polygon is ${\text{2}}{{\text{2}}^0}$, then the measure of exterior angle of that regular polygon will be \[\left( {{{180}^0} - {\text{2}}{{\text{2}}^0}} \right) = {158^0}\]. Clearly, \[{158^0}\] is not a multiple of ${360^0}$. So, it is not possible to have a regular polygon with a measure of each interior angle of ${\text{2}}{{\text{2}}^0}$.

Note- In these types of problems, if the interior angle of the regular polygon is given then it is converted into the exterior angle of the regular polygon. Then using properties of a regular polygon like the sum of all the exterior angles is always equal to ${360^0}$ and each exterior angle is equal, we have to check whether these properties hold true or false. If they hold then that regular polygon is possible else, it is not.

Last updated date: 25th Sep 2023

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