The sum of the interior angles of a hexagon is equal to the sum of six consecutive integers. what is the measure of the smallest interior angle of the hexagon

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Step-by-step explanation:

* Lets explain how to solve the problem

- The consecutive integers are the integers after each other like

  1 , 2 , 3 ,

- The rule of the sum of the interior angles of any polygon is:

  (n - 2) × 180° , where n is the number of the sides of the polygon

- The sum of the interior angles of a hexagon is equal to the sum of

  six consecutive integers

* Lets find the sum of the interior angle of the hexagon

∵ The hexagon has 6 sides

∴ n = 6

∴ The sum of its interior angles = (6 - 2) × 180° = 720°

* Lets find the sum of six consecutive integers

- Assume that the smallest integer is x

∴ The numbers are x , x + 1 , x + 2 , x + 3 , x + 4 , x + 5

∵ Their sum = x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5)

- Add like terms

∴ Their sum = 6x + 15

- Equate the sum of the 6 numbers by the sum of the angles of

 the hexagon

∴ 6x + 15 = 720

- Subtract 15 from both sides

∴ 6x = 705

- Divide both sides by 6

∴ x = 117.5

- But 117 .5 not integer

∴ The sum of the interior angles of a hexagon can not equal the

   sum of six consecutive integers

- But it can be if the numbers are consecutive odd integers

 because the consecutive odd numbers are

 x , x  + 2 , x + 4 , x + 6 , x + 8 , x + 10

∴ Their sum = 6x + 30

∵ 6x + 30 = 720

- Subtract 30 from both sides

∴ 6x = 690

- Divide both sides by 6

∵ x = 115

∵ x represents the measure of the smallest angle

∴ The measure of the smallest interior angle of the hexagon is 115°

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