Question

find the number of sides of the regular polygon where the sum of the interior angles is 2520°.

octagon

16 - gon

24 - gon

hexagon

question 6
2 points
which regular polygon has an exterior angle that has a measure of 9°?

a nonagon

a hexagon

a 40 - gon

a 30 - gon

2 points
question 7
2 points
a pentagon has exterior angles of 71°, 85°, 44°, 2x and 3x. what are the measurements of the final two angles?

8° and 12°

136° and 204°

64° and 96°

32° and 48°

Explanation:

Step1: Use interior angle sum formula

The sum of interior angles of an n-sided polygon is given by $(n-2)\times180^\circ$. Set equal to $2520^\circ$:
$$(n-2)\times180 = 2520$$

Step2: Solve for n

Divide both sides by 180:
$$n-2 = \frac{2520}{180} = 14$$
Add 2 to both sides:
$$n = 14 + 2 = 16$$

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Step1: Use exterior angle property

The sum of exterior angles of any polygon is $360^\circ$. For a regular n-gon, each exterior angle is $\frac{360^\circ}{n}$. Set equal to $9^\circ$:
$$\frac{360}{n} = 9$$

Step2: Solve for n

Rearrange to solve for n:
$$n = \frac{360}{9} = 40$$

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Step1: Sum exterior angles to 360°

Sum all given exterior angles and set equal to $360^\circ$:
$$71 + 85 + 44 + 2x + 3x = 360$$

Step2: Simplify and solve for x

Combine like terms:
$$200 + 5x = 360$$
Subtract 200 from both sides:
$$5x = 160$$
Divide by 5:
$$x = 32$$

Step3: Calculate final two angles

Find $2x$ and $3x$:
$$2x = 2\times32 = 64^\circ$$
$$3x = 3\times32 = 96^\circ$$

Answer:

1. 16-gon
2. A 40-gon
3. 64° and 96°