How to find an angle in a pentagon - Intermediate Geometry

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Question

The angles at 3 verticies of a pentagon are 60, 80 and 100. Which of the following could NOT be the measures of the other 2 angles?

Answer

The sum of the angles in a polygon is

For a pentagon, this equals 540. Since the first 3 angles add up to 240, the remaining 2 angles must add up to $540-240=300^{\circ}$

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Question

What is the sum of three angles in a hexagon if the perimeter of the hexagon is ?

Answer

The perimeter in this question is irrelevant. Use the interior angle formula to determine the total sum of the angles in a hexagon.

$\Theta =(n-2)\cdot180$

There are six interior angles in a hexagon.

$\Theta =(6-2)\cdot180 = 720^{\circ}$

Each angle will be a sixth of the total angle.

$\frac{720}{6}= 120^{\circ}$

Therefore, the sum of three angles in a hexagon is:

$120^{\circ}+120^{\circ}+120^{\circ}=360^{\circ}$

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Question

Add four interior angles in a regular pentagon. What is the result?

Answer

Use the interior angle formula to find the total sum of angles in a pentagon.

$\sum \Theta=(n-2) \cdot 180$

for a pentagon, so substitute this value into the equation and solve:

$\sum \Theta=(5-2) \cdot 180 = 540^{\circ}$

Divide this number by 5, since there are five interior angles.

$\frac{540^{\circ}}{5}=108^{\circ}$

The sum of four interior angles in a regular pentagon is:

$108^{\circ}+108^{\circ}+108^{\circ}+108^{\circ} =432^{\circ}$

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Question

What is the sum of two interior angles of a regular pentagon if the perimeter is 6?

Answer

The perimeter of a regular pentagon has no effect on the interior angles of the pentagon.

Use the following formula to solve for the sum of all interior angles in the pentagon.

$\sum\theta=(n-2)\cdot 180$

Since there are 5 sides in a pentagon, substitute the side length .

$\sum\theta=(5-2)\cdot 180=540$

Divide this by 5 to determine the value of each angle, and then multiply by 2 to determine the sum of 2 interior angles.

$\frac{540}{5}=108$

The sum of 2 interior angles of a pentagon is .

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Question

Let the area of a regular pentagon be . What is the value of an interior angle?

Answer

Area has no effect on the value of the interior angles of a pentagon. To find the sum of all angles of a pentagon, use the following formula, where is the number of sides:

$\sum\theta=(n-2)\cdot 180$

There are 5 sides in a pentagon.

$\sum\theta=(5-2)\cdot 180=540$

Divide this number by 5 to determine the value of each angle.

$\frac{540}{5}=108^\circ$

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Question

Suppose an interior angle of a regular pentagon is . What is ?

Answer

The pentagon has 5 sides. To find the value of the interior angle of a pentagon, use the following formula to find the sum of all interior angles.

$\sum\theta=(n-2)\cdot 180$

Substitute .

$\sum\theta=(5-2)\cdot 180=540$

Divide this number by 5 to determine the value of each interior angle.

$\frac{540}{5}=108$

Every interior angle is 108 degrees. The problem states that an interior angle is . Set these two values equal to each other and solve for .

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Question

True or false: Each of the five angles of a regular pentagon measures $120^{\circ }$ .

Answer

A regular polygon with sides has congruent angles, each of which measures

$\frac{(N-2)180^{\circ }}{N}$

Setting , the common angle measure can be calculated to be

$\frac{(5-2)180^{\circ }}{5} = \frac{3 \cdot 180^{\circ }}{5} = \frac{540^{\circ }}{5} = 108^{\circ }$

The statement is therefore false.

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Question

True or false: Each of the exterior angles of a regular pentagon measures $72 ^{\circ }$ .

Answer

If one exterior angle is taken at each vertex of any polygon, and their measures are added, the sum is $360^{\circ }$ . Each exterior angle of a regular pentagon has the same measure, so if we let be that common measure, then

$5t = 360 ^{\circ }$

Solve for :

$5t \div 5 = 360 ^{\circ } \div 5$

$t = 72 ^{\circ }$

The statement is true.

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Question

Given: Pentagon .

$m \angle P = 108 ^{\circ }$

True, false, or undetermined: Pentagon is regular.

Answer

Suppose Pentagon is regular. Each angle of a regular polygon of sides has measure

$\frac{(N-2)180^{\circ }}{N}$

A pentagon has 5 sides, so set ; each angle of the regular hexagon has measure

$\frac{(5-2)180^{\circ }}{5} = \frac{ 3 \cdot 180^{\circ }}{5} = 108^{\circ }$

Since one angle is given to be of measure $108 ^{\circ }$ , the pentagon might be regular - but without knowing more, it cannot be determined for certain. Therefore, the correct choice is "undetermined".

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