Question

the sum measure of the angles of a regular polygon is 14 right angles. which of the following could be the ratio of the interior angle and extorior angle of the polygon:-\\
7:2
5:3
5:2
7:3

Answer 1

$\begin{gathered}\Large{\bold{\blue{\underline{Formula \: Used \::}}}} \end{gathered}$

$(1). \: \boxed{\pink{\tt \:Sum \: of \: angles = \: (n - 2) \times 180}}$

$(2). \: \boxed{ \pink{\tt \: Exterior \: angle = \dfrac{360}{number \: of \: sides} }}$

$(3). \: \boxed{ \pink{\tt \: Exterior \: angle \: + \: Interior \: angle \: = 180 }}$

$\large\underline\purple{\bold{Solution :- }}$

According to statement,

• The sum measure of the angles of a regular polygon is 14 right angles.

• Let number of sides of regular polygon be 'n'

So,

• We know that

$\rm :\implies\:\tt \:Sum \: of \: angles = \: (n - 2) \times 180$

$\rm :\implies\:14 \times 90 = (n - 2) \times 180$

$\rm :\implies\:7 = n - 2$

$\rm :\implies \boxed{ \blue{ \bf\:n \: = \: 9}}$

So,

• Exterior angle of a regular polygon is given by

$\rm :\implies\:Exterior \: angle \: = \dfrac{360}{n}$

$\rm :\implies\:Exterior \: angle = \dfrac{360}{9}$

$\rm :\implies \boxed{ \red{ \tt{\:Exterior \: angle \: = \: 40}}}$

Now,

• Interior Angle of a regular polygon is given by

$\rm :\implies\:Exterior \: angle + Interior \: angle = 180$

$\rm :\implies\:40 + Interior \: angle = 180$

$\rm :\implies\:Interior \: angle = 180 - 40$

$\rm :\implies \boxed{ \green{ \bf \: \:Interior \: angle \: = \: 140}}$

Hence,

$\rm :\implies\:Interior \: angle : Exterior \: angle$

$\rm :\implies\:140 : 40$

$\bf\implies \: \boxed{ \green{ \bf \: 7 : 2}}$