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Two angles of an eight sided polygon are \[142^\circ \]land \[176^\circ \]. If the remaining angles equal to each other; find the magnitude of each of the angles.
A. \[156^\circ \]
B. \[127^\circ \]
C. \[136^\circ \]
D. \[116^\circ \]

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Last updated date: 20th Jun 2024
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Answer
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Hint: To solve the question at first we have to find the sum of all interior angles of an eight sided polygon using the angle formula of a polygon. Then we have to consider the magnitude of each of the remaining angles to be\[{x^ \circ }\]. After that we must add all of the eight angles and equate to the sum of the interior angle obtained using the formula before. Finally we can get the value of x by solving the equation.

Complete step-by-step answer:
We know the formula that the sum of all angles of a polygon is given by \[{180^ \circ }\left( {n - 2} \right)\] where n is the number of sides of the polygon.
Here \[n = 8\], hence by applying the formula, the sum of all interior angles of the eight sided polygon is given by,\[180^\circ \left( {8 - 2} \right) = 180^\circ \times 6 = 1080^\circ \].
Given that the two angles are \[142^\circ \]and\[176^\circ \]. Let the remaining six angles are each of magnitude \[x^\circ \]. Then the sum of the angles is \[142^\circ + 176^\circ + 6x = 318^\circ + 6x\].
Therefore we get,
\[\begin{align}
\Rightarrow 318^\circ + 6x = 1080^\circ
\Rightarrow 6x = 762^\circ \\
\Rightarrow x = 127^\circ \\
\end{align} \]
Hence we got that the remaining angles are each of magnitude \[127^\circ \].

Note: The sum of all interior angles of an eight sided polygon is \[1080{}^\circ \]. Each of the angles of an eight sided angle must be obtuse. Always remember that, sum of all angles of a polygon is given by \[{{180}^{\circ }}\left( n-2 \right)\] where n is the number of sides of the polygon. When you do calculation, then try not to do any calculation error as this will change the final answer.