Answer
420.6k+ views
Hint: To solve this question, we will first calculate the sum of the interior angle by writing them as 6x + 114 as 6 angles are of measure x and one of 114. Then we will finally equate it to the formula of the sum of the interior angles of n sides polygon as \[\left( 2n-4 \right)\times 90\] to get the result.
Complete step-by-step solution
We are given that one angle of a seven-sided polygon is 114. A seven-sided polygon looks like the figure below.
The 7 sides are also mentioned in the figure given that each of the six angles is x degrees. Therefore, the angles are as:
There is a formula of the sum of the interior angles of an n sided polygon given as \[\left( 2n-4 \right)\times 90....\left( i \right)\]
Now, the given one angle is 114. The other 6 sides measure x. Therefore, the sum of all interior angles is \[6x+114.....\left( ii \right)\]
Using the formula stated above in the equation (i) and equating to the equation (ii), we get,
\[6x+114=\left( 2n-4 \right)\times 90\]
Using n = 7, we have,
\[\Rightarrow 6x+114=\left( 2\times 7-4 \right)\times 90\]
Simplifying the bracket as 14 - 4 = 10 and taking 114 to the RHS, we have
\[\Rightarrow 6x=10\times 90-114\]
\[\Rightarrow 6x=900-114\]
\[\Rightarrow 6x=786\]
\[\Rightarrow x={{131}^{\circ }}\]
Therefore, the value of x is 131 degrees. Hence, the option (c) is the right answer.
Note: A possibility of mistake in this question is directly equating \[7{{x}^{\circ }}\] to \[\left( 2n-4 \right)\times {{90}^{\circ }}.\] This would be wrong as one measure of the angle is already given as \[{{114}^{\circ }}.\] We only need 6x and not 7x and hence \[6x+114=\left( 2n-4 \right)\times {{90}^{\circ }}\] and not the above written.
Complete step-by-step solution
We are given that one angle of a seven-sided polygon is 114. A seven-sided polygon looks like the figure below.
![seo images](https://www.vedantu.com/question-sets/0a73d18f-b8c8-48f1-bddb-952284f06ae41790830608026173024.png)
The 7 sides are also mentioned in the figure given that each of the six angles is x degrees. Therefore, the angles are as:
![seo images](https://www.vedantu.com/question-sets/5a93f2e8-ca74-4e2a-a8f4-f672fb75fb815992986094408522185.png)
There is a formula of the sum of the interior angles of an n sided polygon given as \[\left( 2n-4 \right)\times 90....\left( i \right)\]
Now, the given one angle is 114. The other 6 sides measure x. Therefore, the sum of all interior angles is \[6x+114.....\left( ii \right)\]
Using the formula stated above in the equation (i) and equating to the equation (ii), we get,
\[6x+114=\left( 2n-4 \right)\times 90\]
Using n = 7, we have,
\[\Rightarrow 6x+114=\left( 2\times 7-4 \right)\times 90\]
Simplifying the bracket as 14 - 4 = 10 and taking 114 to the RHS, we have
\[\Rightarrow 6x=10\times 90-114\]
\[\Rightarrow 6x=900-114\]
\[\Rightarrow 6x=786\]
\[\Rightarrow x={{131}^{\circ }}\]
Therefore, the value of x is 131 degrees. Hence, the option (c) is the right answer.
Note: A possibility of mistake in this question is directly equating \[7{{x}^{\circ }}\] to \[\left( 2n-4 \right)\times {{90}^{\circ }}.\] This would be wrong as one measure of the angle is already given as \[{{114}^{\circ }}.\] We only need 6x and not 7x and hence \[6x+114=\left( 2n-4 \right)\times {{90}^{\circ }}\] and not the above written.
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