Intermediate Geometry - Intermediate Geometry

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Question

What is the measure of one exterior angle of a regular twenty-sided polygon?

Answer

The sum of the exterior angles of any polygon, one at each vertex, is $360^{\circ }$ . In a regular polygon, the exterior angles all have the same measure, so divide 360 by the number of angles, which, here, is 20, the same as the number of sides.

$360\div 20 = 18$

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Question

Which of the following cannot be the six interior angle measures of a hexagon?

Answer

The sum of the interior angle measures of a hexagon is $180 (6-2)=720^{\circ }$

Add the angle measures in each group.

In each case, the angle measures add up to 720, so the answer is that all of these can be the six interior angle measures of a hexagon.

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Question

There is a regular hexagon with a side length of . What is the measure of an internal angle?

Answer

Given that the hexagon is a regular hexagon, this means that all the side length are congruent and all internal angles are congruent. The question requires us to solve for the measure of an internal angle. Given the aforementioned definition of a regular polygon, this means that there must only be one correct answer.

In order to solve for the answer, the question provides additional information that isn't necessarily required. The measure of an internal angle can be solved for using the equation:

$\frac{180^{\circ}(n-2)}{n}$ where is the number of sides of the polygon.

In this case, .

For this problem, the information about the side length may be negated.

$\frac{180^{\circ}(6-2)}{6}$

$\frac{180^{\circ}(4)}{6}$

$\frac{180^{\circ}(2)}{3}$

$120^{\circ}$

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Question

What is the interior angle of a regular hexagon if the area is 15?

Answer

The area has no relevance to find the angle of a regular hexagon.

There are 6 sides in a regular hexagon. Use the following formula to determine the interior angle.

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

Substitute sides to determine the sum of all interior angles of the hexagon in degrees.

$\sum\theta=(6-2)\cdot 180=720$

Since there are 6 sides, divide this number by 6 to determine the value of each interior angle.

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

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Question

Given: Regular Hexagon with center . Construct segments $\overline{CA}$ and $\overline{CO}$ to form Quadrilateral .

True or false: Quadrilateral is a rectangle.

Answer

Below is regular Hexagon with center , a segment drawn from to each vertex - that is, each of its radii drawn.

Each angle of a regular hexagon measures $120^{\circ }$ ; by symmetry, each radius bisects an angle of the hexagon, so

$m \angle CAG = 60 ^{\circ }$ .

The angles of a rectangle must measure $90^{\circ }$ , so it has been disproved that Quadrilateral is a rectangle.

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Question

True or false: Each of the six angles of a regular hexagon 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

$\frac{(6-2)180^{\circ }}{6} = \frac{4 \cdot 180^{\circ }}{6} = \frac{720^{\circ }}{6} = 120^{\circ }$

The statement is therefore true.

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Question

True or false: Each of the exterior angles of a regular hexagon measures $80^{\circ }$ .

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 hexagon has the same measure, so if we let be that common measure, then

$6t = 360 ^{\circ }$

Solve for :

$6t \div 6 = 360 ^{\circ } \div 6$

$t = 60^{\circ }$

The statement is false.

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Question

Given: Hexagon .

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

True, false, or undetermined: Hexagon is regular.

Answer

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

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

A hexagon has 6 sides, so set ; each angle of the regular hexagon has measure

$\frac{(6-2)180^{\circ }}{6} = \frac{ 4 \cdot 180^{\circ }}{6} = 120^{\circ }$

Since one angle is given to be of measure $108 ^{\circ }$ , the hexagon cannot be regular.

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Question

A parallelogram contains 2 angles measuring 135 and 45. What are the measures of the other 2 angles?

Answer

Parallelograms have angles totalling 360 degrees, but also have matching pairs of angles at the ends of diagonals. Therefore the 2 additional angles must match the 2 given in the question.

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Question

Using the above rhombus, find the measurement of angle

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees)--i.e. angles degrees.

Thus, the solution is:

$\measuredangle D=180$

$\measuredangle D=180-110=70$

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Question

Using the above rhombus, find the measurement of angle .

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees)--i.e. angles degrees.

Thus, $\measuredangle C=\measuredangle A$

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Question

Using the above rhombus, find the measurement of angle

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Thus, the solution is:

$\measuredangle H+\measuredangle G=180^\circ$
$47+\measuredangle G=180^\circ$
$\measuredangle G=180-47=133^\circ$

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Question

Using the above rhombus, find the sum of angle and angle .

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Thus, the solution is:

$\measuredangle H=47^\circ$

$\measuredangle F=47^\circ$

$47^\circ+47^\circ=94^\circ$

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Question

Given that the measurement of angle degrees, find the sum of angle and angle

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees)--i.e. angles degrees.

The solution to this problem is:

$\measuredangle B+\measuredangle C=180^\circ$

$52+\measuredangle C=180^\circ$

$C= 180-52=128^\circ$

$\measuredangle C=\measuredangle A$

Therefore,

$128+128=256^\circ$

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Question

In the above rhombus, angle has a measurement of degrees. Find the sum of angles and

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees)--i.e. angles degrees.

The solution to this problem is:

$\measuredangle D=\measuredangle B$

Thus,

$52+52=104^\circ$

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Question

Using the parallelogram above, find the measurement of angle

Answer

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Since, angle and are supplementary the solution is:

$180-99=81^\circ$

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Question

Using the parallelogram above, find the sum of angles and .

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

The first step to solving this problem is to find the measurement of angle . Since angle is a supplementary angle to angle , angle

$180-99=81^\circ$

Since, angle and are opposite interior angles they must be equivalent.

Thus, the final solution is:

$81+81=162^\circ$

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Question

Using the parallelogram above, find the sum of angles and .

Answer

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Since, angles and are opposite interior angles, they must be equivalent.

Therefore, the solution is:

$99+99=198^\circ$

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Question

In the parallelogram shown above, angle is degrees. Find the measure of angle

Answer

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Since, angles and are opposite interior angles, thus they must be equivalent.

$\measuredangle E=72^\circ$ , therefore $\measuredangle G=72^\circ$

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Question

In the parallelogram shown above, angle is degrees. Find the sum of angles and .

Answer

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Thus, the solution is:

$180-72=108^\circ$

Since both angles and equal There sum must equal $108+108=216^\circ$

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