Question

!50 POINTS! (3 SIMPLE GEOMETRY QUESTION)

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Answer 1

1) 9

2) 135

3) 7

Explanation:

The sum of interior angles of a polygon with n sides is: (n - 2)*180

1)

`(n - 2)*180 = 1260\\\\\implies n-2 = (1260)/(180)\\\\\implies n-2=7\\\\\implies n = 7+2`

⇒ n = 9

2) n = 8

The sum of interior angles is: (n - 2)*180

⇒ Each angle measures:
`((n - 2)*180)/(n)`

`= ((8 - 2)*180)/(8)\\\\= (6*180)/(8)\\\\= (1080)/(8)\\\\=135`

3)

`(n - 2)*180 = 900\\\\\implies n-2 = (900)/(180)\\\\\implies n-2=5\\\\\implies n = 5+2`

⇒ n = 7

Answer 2

1) 9 sides

2) 135°

3) 7 sides

Explanation:

Question 1

To find the number of sides of a polygon, given the sum of its interior angles, we can use the formula:

`\boxed{S= 180^(\circ)(n-2)}`

where:

• S is the sum of the interior angles.
• n is the the number of sides of the polygon.

Given that the sum of the measures of the interior angles is 1260° substitute S = 1260° into the formula and solve for n:

`\begin{aligned}180^(\circ)(n - 2)&=1260^(\circ)\\\\(180^(\circ)(n - 2))/(180^(\circ))&=(1260^(\circ))/(180^(\circ))\\\\n-2&=7\\\\n-2+2&=7+2\\\\n&=9\end{aligned}`

Therefore, the polygon has 9 sides.

`\hrulefill`

Question 2

To find the measure of each interior angle of a regular octagon, we can use the formula:

`\boxed{\textsf{Interior angle of a regular polygon} = (180^(\circ)(n-2))/(n)}`

where n is the number of sides of the polygon.

The number of sides of a regular octagon is 8. Therefore, substitute n = 8 into the formula:

`\begin{aligned}{\textsf{Interior angle of a regular octagon} &= (180^(\circ)(8-2))/(8)}\\\\&= (180^(\circ)(6))/(8)}\\\\&= (1080^(\circ))/(8)}\\\\&=135^(\circ) \end{aligned}`

Therefore, the measure of each interior angle of a regular octagon is 135°.

`\hrulefill`

Question 3

To find the number of sides of a polygon given the sum of its interior angles, we can use the formula:

`\boxed{S= 180^(\circ)(n-2)}`

where:

• S is the sum of the interior angles.
• n is the the number of sides of the polygon.

Given that the sum of the measures of the interior angles is 900°, substitute S = 900° into the formula and solve for n:

`\begin{aligned}900^(\circ)&= 180^(\circ)(n-2)}\\\\(900^(\circ))/(180^(\circ))&= (180^(\circ)(n-2))/(180^(\circ))\\\\5&= n-2\\\\5+2&=n-2+2\\\\7&=n\\\\n&=7\end{aligned}`

Therefore, the polygon has 7 sides.