Calculating an angle in a polygon - GMAT Quantitative
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What is the measure of one exterior angle of a regular twenty-four sided polygon?
What is the measure of one exterior angle of a regular twenty-four sided polygon?
The sum of the measures of the exterior angles of any polygon, one at each vertex, is
. Since a regular polygon with twenty-four sides has twenty-four congruent angles, and therefore, congruent exterior angles, just divide:

The sum of the measures of the exterior angles of any polygon, one at each vertex, is . Since a regular polygon with twenty-four sides has twenty-four congruent angles, and therefore, congruent exterior angles, just divide:
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Which of the following figures would have exterior angles none of whose degree measures is an integer?
Which of the following figures would have exterior angles none of whose degree measures is an integer?
The sum of the degree measures of any polygon is
. A regular polygon with
sides has exterior angles of degree measure
. For this to be an integer, 360 must be divisible by
.
We can test each of our choices to see which one fails this test.





Only the eighty-sided regular polygon fails this test, making this the correct choice.
The sum of the degree measures of any polygon is . A regular polygon with
sides has exterior angles of degree measure
. For this to be an integer, 360 must be divisible by
.
We can test each of our choices to see which one fails this test.
Only the eighty-sided regular polygon fails this test, making this the correct choice.
Compare your answer with the correct one above
You are given Pentagon
such that:

and

Calculate 
You are given Pentagon such that:
and
Calculate
Let
be the common measure of
,
,
, and 
Then

The sum of the measures of the angles of a pentagon is
degrees; this translates to the equation

or




Let be the common measure of
,
,
, and
Then
The sum of the measures of the angles of a pentagon is degrees; this translates to the equation
or
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The above diagram shows a regular pentagon and a regular hexagon sharing a side. What is the measure of
?
The above diagram shows a regular pentagon and a regular hexagon sharing a side. What is the measure of ?
The measure of each interior angle of a regular pentagon is

The measure of each interior angle of a regular hexagon is

The measure of
is the difference of the two, or
.
The measure of each interior angle of a regular pentagon is
The measure of each interior angle of a regular hexagon is
The measure of is the difference of the two, or
.
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The above diagram shows a regular pentagon and a regular hexagon sharing a side. Give
.
The above diagram shows a regular pentagon and a regular hexagon sharing a side. Give .
This can more easily be explained if the shared side is extended in one direction, and the new angles labeled.

and
are exterior angles of the regular polygons. Also, the measures of the exterior angles of any polygon, one at each vertex, total
. Therefore,


Add the measures of the angles to get
:

This can more easily be explained if the shared side is extended in one direction, and the new angles labeled.
and
are exterior angles of the regular polygons. Also, the measures of the exterior angles of any polygon, one at each vertex, total
. Therefore,
Add the measures of the angles to get :
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Note: Figure NOT drawn to scale
The figure above shows a square inside a regular pentagon. Give
.
Note: Figure NOT drawn to scale
The figure above shows a square inside a regular pentagon. Give .
Each angle of a square measures
; each angle of a regular pentagon measures
. To get
, subtract:
.
Each angle of a square measures ; each angle of a regular pentagon measures
. To get
, subtract:
.
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Which of the following cannot be the measure of an exterior angle of a regular polygon?
Which of the following cannot be the measure of an exterior angle of a regular polygon?
The sum of the measures of the exterior angles of any polygon, one per vertex, is
. In a regular polygon of
sides , then all
of these exterior angles are congruent, each measuring
.
If
is the measure of one of these angles, then
, or, equivalently,
. Therefore, for
to be a possible measure of an exterior angle, it must divide evenly into 360. We divide each in turn:




Since 16 is the only one of the choices that does not divide evenly into 360, it cannot be the measure of an exterior angle of a regular polygon.
The sum of the measures of the exterior angles of any polygon, one per vertex, is . In a regular polygon of
sides , then all
of these exterior angles are congruent, each measuring
.
If is the measure of one of these angles, then
, or, equivalently,
. Therefore, for
to be a possible measure of an exterior angle, it must divide evenly into 360. We divide each in turn:
Since 16 is the only one of the choices that does not divide evenly into 360, it cannot be the measure of an exterior angle of a regular polygon.
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Note: Figure NOT drawn to scale.
Given:



Evaluate
.
Note: Figure NOT drawn to scale.
Given:
Evaluate .
Call
the measure of 

, and 
so

The sum of the measures of the angles of a hexagon is
, so




, which is the measure of
.
Call the measure of
, and
so
The sum of the measures of the angles of a hexagon is , so
, which is the measure of
.
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What is the arithmetic mean of the measures of the angles of a nonagon (a nine-sided polygon)?
What is the arithmetic mean of the measures of the angles of a nonagon (a nine-sided polygon)?
The sum of the measures of the nine angles of any nonagon is calculated as follows:

Divide this number by nine to get the arithmetic mean of the measures:

The sum of the measures of the nine angles of any nonagon is calculated as follows:
Divide this number by nine to get the arithmetic mean of the measures:
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What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?
What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?
The sum of the measures of the nine angles of any nonagon is calculated as follows:

The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.
Case 1: Each angle measures
.
The set is
and the median is 140.
Case 2: Eight of the angles measure
and one of them measures
.
The set is
and the median is 139.
In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.
The sum of the measures of the nine angles of any nonagon is calculated as follows:
The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.
Case 1: Each angle measures .
The set is and the median is 140.
Case 2: Eight of the angles measure and one of them measures
.
The set is and the median is 139.
In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.
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You are given a quadrilateral and a pentagon. What is the mean of the measures of the interior angles of the two polygons?
You are given a quadrilateral and a pentagon. What is the mean of the measures of the interior angles of the two polygons?
The mean of the measures of the four angles of the quadrilateral and the five angles of of the pentagon is their sum divided by 9.
The sum of the measures of the interior angles of any quadrilateral is
. The sum of the measures of the interior angles of any pentagon is
.
The sum of the measures of the interior angles of both polygons is therefore
. Divide by 9:

The mean of the measures of the four angles of the quadrilateral and the five angles of of the pentagon is their sum divided by 9.
The sum of the measures of the interior angles of any quadrilateral is . The sum of the measures of the interior angles of any pentagon is
.
The sum of the measures of the interior angles of both polygons is therefore . Divide by 9:
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Note: Figure NOT drawn to scale.
Given Regular Pentagon
. What is
?
Note: Figure NOT drawn to scale.
Given Regular Pentagon . What is
?
Quadrilateral
is a trapezoid, so
.
, so



Quadrilateral is a trapezoid, so
.
, so
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The angles of a pentagon measure
.
Evaluate
.
The angles of a pentagon measure .
Evaluate .
The sum of the degree measures of the angles of a (five-sided) pentagon is
, so we can set up and solve the equation:






The sum of the degree measures of the angles of a (five-sided) pentagon is , so we can set up and solve the equation:
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The measures of the angles of a pentagon are: 
What is
equal to?
The measures of the angles of a pentagon are:
What is equal to?
The degree measures of the interior angles of a pentagon total
, so






The degree measures of the interior angles of a pentagon total , so
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What is the measure of an angle in a regular octagon?
What is the measure of an angle in a regular octagon?
On octagon has
sides. The word regular means that all of the angles are equal. Therefore, we can use the general equation for finding the angle measurement of a regular polygon:
, where
is the number of sides of the polygon.
.
On octagon has sides. The word regular means that all of the angles are equal. Therefore, we can use the general equation for finding the angle measurement of a regular polygon:
, where
is the number of sides of the polygon.
.
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What is the measure of one exterior angle of a regular twenty-four sided polygon?
What is the measure of one exterior angle of a regular twenty-four sided polygon?
The sum of the measures of the exterior angles of any polygon, one at each vertex, is
. Since a regular polygon with twenty-four sides has twenty-four congruent angles, and therefore, congruent exterior angles, just divide:

The sum of the measures of the exterior angles of any polygon, one at each vertex, is . Since a regular polygon with twenty-four sides has twenty-four congruent angles, and therefore, congruent exterior angles, just divide:
Compare your answer with the correct one above
Which of the following figures would have exterior angles none of whose degree measures is an integer?
Which of the following figures would have exterior angles none of whose degree measures is an integer?
The sum of the degree measures of any polygon is
. A regular polygon with
sides has exterior angles of degree measure
. For this to be an integer, 360 must be divisible by
.
We can test each of our choices to see which one fails this test.





Only the eighty-sided regular polygon fails this test, making this the correct choice.
The sum of the degree measures of any polygon is . A regular polygon with
sides has exterior angles of degree measure
. For this to be an integer, 360 must be divisible by
.
We can test each of our choices to see which one fails this test.
Only the eighty-sided regular polygon fails this test, making this the correct choice.
Compare your answer with the correct one above
You are given Pentagon
such that:

and

Calculate 
You are given Pentagon such that:
and
Calculate
Let
be the common measure of
,
,
, and 
Then

The sum of the measures of the angles of a pentagon is
degrees; this translates to the equation

or




Let be the common measure of
,
,
, and
Then
The sum of the measures of the angles of a pentagon is degrees; this translates to the equation
or
Compare your answer with the correct one above

The above diagram shows a regular pentagon and a regular hexagon sharing a side. What is the measure of
?
The above diagram shows a regular pentagon and a regular hexagon sharing a side. What is the measure of ?
The measure of each interior angle of a regular pentagon is

The measure of each interior angle of a regular hexagon is

The measure of
is the difference of the two, or
.
The measure of each interior angle of a regular pentagon is
The measure of each interior angle of a regular hexagon is
The measure of is the difference of the two, or
.
Compare your answer with the correct one above

The above diagram shows a regular pentagon and a regular hexagon sharing a side. Give
.
The above diagram shows a regular pentagon and a regular hexagon sharing a side. Give .
This can more easily be explained if the shared side is extended in one direction, and the new angles labeled.

and
are exterior angles of the regular polygons. Also, the measures of the exterior angles of any polygon, one at each vertex, total
. Therefore,


Add the measures of the angles to get
:

This can more easily be explained if the shared side is extended in one direction, and the new angles labeled.
and
are exterior angles of the regular polygons. Also, the measures of the exterior angles of any polygon, one at each vertex, total
. Therefore,
Add the measures of the angles to get :
Compare your answer with the correct one above