Question

# Three angles of a seven-sided polygon are ${{132}^{o}}$ each and the remaining four angles are equal. Find the value of each equal angle.(a) ${{125}^{o}}$(b) ${{156}^{o}}$(c) ${{126}^{o}}$(d) ${{130}^{o}}$

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.

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.