Answer

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Hint: First of all draw the diagram of the 7 sided polygon that is heptagon to get a clear idea about the question. Now, use the formula to get the sum of all the interior angles \[=\left( n-2 \right)\times {{180}^{o}}\] where n is the number of sides of the polygon. Then substitute the value of the given angles to find the remaining angles.

Complete step-by-step answer:

Here we are given that the three angles of the seven-sided polygon are \[{{132}^{o}}\] each and the remaining four angles are equal. We have to find the value of each equal angle.

First of all, let us draw the 7 sided polygon which is also known as heptagon.

Here polygon ABCDEFG is our heptagon with 7 interior angles \[\angle A,\angle B,\angle C,\angle D,\angle E,\angle F\text{ and }\angle G\].

We know that for any n sided polygon, the sum of its interior angles \[=\left( n-2 \right)\times {{180}^{o}}\].

Therefore, for a heptagon which is a 7 sided polygon that is n = 7. We get,

Sum of the interior angles \[=\left( 7-2 \right)\times {{180}^{o}}\]

\[=5\times {{180}^{o}}\]

\[={{900}^{o}}\]

Now, we are given that the three angles of this heptagon are \[{{132}^{o}}\] each.

So, let us assume that \[\angle A=\angle B=\angle C={{132}^{o}}\]

Also, we are given that the remaining four angles of this heptagon are equal.

So, let us assume \[\angle D=\angle E=\angle F=\angle G=\theta \]

We know that the sum of all the interior angles is \[{{900}^{o}}\] in this heptagon. So, we get,

\[\angle A+\angle B+\angle C+\angle D+\angle E+\angle F+\angle G={{900}^{o}}\]

By substituting \[\angle A=\angle B=\angle C={{132}^{o}}\] and \[\angle D=\angle E=\angle F=\angle G=\theta \], we get,

\[\Rightarrow {{132}^{o}}+{{132}^{o}}+{{132}^{o}}+\theta +\theta +\theta +\theta ={{900}^{o}}\]

Or, \[\left( {{132}^{o}} \right)\times 3+4\theta ={{900}^{o}}\]

\[\Rightarrow {{396}^{o}}+4\theta ={{900}^{o}}\]

By subtracting \[{{396}^{o}}\] on both the sides, we get,

\[\Rightarrow 4\theta ={{900}^{o}}-{{396}^{o}}\]

\[\Rightarrow 4\theta ={{504}^{o}}\]

By dividing 4 on both the sides, we get,

\[\theta =\dfrac{{{504}^{o}}}{4}={{126}^{o}}\]

Hence, we get the value of each equal angle \[={{126}^{o}}\].

Therefore, option (c) is the right option.

Note: Here students can verify their answer as follows:

As we know that the sum of all the interior angles of the heptagon is \[{{900}^{o}}\]. So, here we will verify if the sum of all the interior angles is coming \[{{900}^{o}}\] or by individually adding them.

So, \[\angle A+\angle B+\angle C+\angle D+\angle E+\angle F+\angle G={{900}^{o}}\]

By substituting \[\angle A=\angle B=\angle C={{132}^{o}}\] and \[\angle D=\angle E=\angle F=\angle G={{126}^{o}}\] in the above equation, we get,

\[\left( {{132}^{o}}+{{132}^{o}}+{{132}^{o}} \right)+\left( {{126}^{o}}+{{126}^{o}}+{{126}^{o}}+{{126}^{o}} \right)={{900}^{o}}\]

\[=\left( {{132}^{o}}\times 3 \right)+\left( {{126}^{o}}\times 4 \right)={{900}^{o}}\]

\[\Rightarrow {{396}^{o}}+{{504}^{o}}={{900}^{o}}\]

\[\Rightarrow {{900}^{o}}={{900}^{o}}\]

Hence, we get LHS = RHS.

This means that our answer is correct.

Complete step-by-step answer:

Here we are given that the three angles of the seven-sided polygon are \[{{132}^{o}}\] each and the remaining four angles are equal. We have to find the value of each equal angle.

First of all, let us draw the 7 sided polygon which is also known as heptagon.

Here polygon ABCDEFG is our heptagon with 7 interior angles \[\angle A,\angle B,\angle C,\angle D,\angle E,\angle F\text{ and }\angle G\].

We know that for any n sided polygon, the sum of its interior angles \[=\left( n-2 \right)\times {{180}^{o}}\].

Therefore, for a heptagon which is a 7 sided polygon that is n = 7. We get,

Sum of the interior angles \[=\left( 7-2 \right)\times {{180}^{o}}\]

\[=5\times {{180}^{o}}\]

\[={{900}^{o}}\]

Now, we are given that the three angles of this heptagon are \[{{132}^{o}}\] each.

So, let us assume that \[\angle A=\angle B=\angle C={{132}^{o}}\]

Also, we are given that the remaining four angles of this heptagon are equal.

So, let us assume \[\angle D=\angle E=\angle F=\angle G=\theta \]

We know that the sum of all the interior angles is \[{{900}^{o}}\] in this heptagon. So, we get,

\[\angle A+\angle B+\angle C+\angle D+\angle E+\angle F+\angle G={{900}^{o}}\]

By substituting \[\angle A=\angle B=\angle C={{132}^{o}}\] and \[\angle D=\angle E=\angle F=\angle G=\theta \], we get,

\[\Rightarrow {{132}^{o}}+{{132}^{o}}+{{132}^{o}}+\theta +\theta +\theta +\theta ={{900}^{o}}\]

Or, \[\left( {{132}^{o}} \right)\times 3+4\theta ={{900}^{o}}\]

\[\Rightarrow {{396}^{o}}+4\theta ={{900}^{o}}\]

By subtracting \[{{396}^{o}}\] on both the sides, we get,

\[\Rightarrow 4\theta ={{900}^{o}}-{{396}^{o}}\]

\[\Rightarrow 4\theta ={{504}^{o}}\]

By dividing 4 on both the sides, we get,

\[\theta =\dfrac{{{504}^{o}}}{4}={{126}^{o}}\]

Hence, we get the value of each equal angle \[={{126}^{o}}\].

Therefore, option (c) is the right option.

Note: Here students can verify their answer as follows:

As we know that the sum of all the interior angles of the heptagon is \[{{900}^{o}}\]. So, here we will verify if the sum of all the interior angles is coming \[{{900}^{o}}\] or by individually adding them.

So, \[\angle A+\angle B+\angle C+\angle D+\angle E+\angle F+\angle G={{900}^{o}}\]

By substituting \[\angle A=\angle B=\angle C={{132}^{o}}\] and \[\angle D=\angle E=\angle F=\angle G={{126}^{o}}\] in the above equation, we get,

\[\left( {{132}^{o}}+{{132}^{o}}+{{132}^{o}} \right)+\left( {{126}^{o}}+{{126}^{o}}+{{126}^{o}}+{{126}^{o}} \right)={{900}^{o}}\]

\[=\left( {{132}^{o}}\times 3 \right)+\left( {{126}^{o}}\times 4 \right)={{900}^{o}}\]

\[\Rightarrow {{396}^{o}}+{{504}^{o}}={{900}^{o}}\]

\[\Rightarrow {{900}^{o}}={{900}^{o}}\]

Hence, we get LHS = RHS.

This means that our answer is correct.

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