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Mathematics

Is it possible to have a polygon whose sum of interior angles is:

(i) 870°

(ii) 2340°

(iii) 7 right angles ?

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Answer

(i) 870°

According to the properties of a polygon, if there are n sides, then the sum of its interior angles is (2n - 4) x 90°.

So,

(2n - 4) x 90° = 870°

⇒ (2n - 4) = 870°90°\dfrac{870°}{90°}

⇒ 2n - 4 = 9.66……..

⇒ 2n = 9.66…. + 4

⇒ 2n = 13.66….

Since 2n must be an integer, it is not possible for 2n to equal 13.666….

Hence, it is not possible to have a polygon with a sum of interior angles equal to 870°.

(ii) 2340°

According to the properties of a polygon, if there are n sides, then the sum of its interior angles is (2n - 4) x 90°.

So,

(2n - 4) x 90° = 2340°

⇒ (2n - 4) = 2340°90°\dfrac{2340°}{90°}

⇒ 2n - 4 = 26

⇒ 2n = 26 + 4

⇒ 2n = 30

⇒ n = 302\dfrac{30}{2}

⇒ n = 15

Hence, it is possible to have a polygon whose sum of interior angles is 2340°.

(iii) 7 right angles

According to the properties of a polygon, if there are n sides, then the sum of its interior angles is (2n - 4) x 90°.

Given that the sum of its interior angles is 7 right angles:

7 x 90° = 630°

So,

(2n - 4) x 90° = 630°

⇒ (2n - 4) = 630°90°\dfrac{630°}{90°}

⇒ 2n - 4 = 7

⇒ 2n = 7 + 4

⇒ 2n = 11

Since 2n must be an integer, 2n = 11 is not possible.

Hence, it is not possible to have a polygon whose sum of interior angles is 630°.

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