Geometry - GMAT Quantitative
Card 1 of 6760
What is the measurement of an angle that is supplementary to a
angle?
What is the measurement of an angle that is supplementary to a angle?
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Two angles are supplementary if the total of their degree measures is
. Therefore, an angle supplementary to a
angle measures
.
Two angles are supplementary if the total of their degree measures is . Therefore, an angle supplementary to a
angle measures
.
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What is the measure of an angle that is complementary to a
angle?
What is the measure of an angle that is complementary to a angle?
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Two angles are complementary if the total of their degree measures is
. Therefore, an angle complementary to a
angle measures
.
Two angles are complementary if the total of their degree measures is . Therefore, an angle complementary to a
angle measures
.
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Find the equation of the line that is perpendicular to the line connecting the points
.
Find the equation of the line that is perpendicular to the line connecting the points .
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Lines are perpendicular if their slopes are negative reciprocals of each other. First we need to find the slope of the line in the question stem.

The negative reciprocal of 3 is
, so our answer will have a slope of
. Let's go through the answer choices and see.
: This line is of the form
, where
is the slope. The slope is 3, so this line is parallel, not perpendicular, to our line in question.
: The slope here is
, also wrong.
: The slope of this line is
. This is the reciprocal, but not the negative reciprocal, so this is also incorrect.
The line between the points
:
.
This is the correct answer! Let's check the last answer choice as well.
The line between points
:
, which is incorrect.
Lines are perpendicular if their slopes are negative reciprocals of each other. First we need to find the slope of the line in the question stem.
The negative reciprocal of 3 is , so our answer will have a slope of
. Let's go through the answer choices and see.
: This line is of the form
, where
is the slope. The slope is 3, so this line is parallel, not perpendicular, to our line in question.
: The slope here is
, also wrong.
: The slope of this line is
. This is the reciprocal, but not the negative reciprocal, so this is also incorrect.
The line between the points :
.
This is the correct answer! Let's check the last answer choice as well.
The line between points :
, which is incorrect.
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Determine whether the lines with equations
and
are perpendicular.
Determine whether the lines with equations and
are perpendicular.
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If two equations are perpendicular, then they will have inverse negative slopes of each other. So if we compare the slopes of the two equations, then we can find the answer. For the first equation we have 
so the slope is
.
So for the equations to be perpendicular, the other equation needs to have a slope of 3. For the second equation, we have

so the slope is
.
Since the slope of the second equation is not equal to 3, then the lines are not perpendicular.
If two equations are perpendicular, then they will have inverse negative slopes of each other. So if we compare the slopes of the two equations, then we can find the answer. For the first equation we have
so the slope is .
So for the equations to be perpendicular, the other equation needs to have a slope of 3. For the second equation, we have
so the slope is .
Since the slope of the second equation is not equal to 3, then the lines are not perpendicular.
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Figure NOT drawn to scale.
Refer to the above figure.
True or false: 
Statement 1:
is a right angle.
Statement 2:
and
are supplementary.

Figure NOT drawn to scale.
Refer to the above figure.
True or false:
Statement 1: is a right angle.
Statement 2: and
are supplementary.
Tap to reveal answer
Statement 1 alone establishes by definition that
, but does not establish any relationship between
and
.
By Statement 2 alone, since same-side interior angles are supplementary,
, but no conclusion can be drawn about the relationship of
, since the actual measures of the angles are not given.
Assume both statements are true. If two lines are parallel, then any line in their plane perpendicular to one must be perpendicular to the other.
and
, so it can be established that
.
Statement 1 alone establishes by definition that , but does not establish any relationship between
and
.
By Statement 2 alone, since same-side interior angles are supplementary, , but no conclusion can be drawn about the relationship of
, since the actual measures of the angles are not given.
Assume both statements are true. If two lines are parallel, then any line in their plane perpendicular to one must be perpendicular to the other. and
, so it can be established that
.
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Refer to the above figure.
. True or false: 
Statement 1: 
Statement 2:
and
are supplementary.

Refer to the above figure. . True or false:
Statement 1:
Statement 2: and
are supplementary.
Tap to reveal answer
If transversal
crosses two parallel lines
and
, then same-side interior angles are supplementary, so
and
are supplementary angles. Also, corresponding angles are congruent, so
.
By Statement 1 alone, angles
and
are congruent as well as supplementary; by Statement 2 alone,
and
are also supplementary as well as congruent. Two angles that are both supplementary and congruent are both right angles, so from either statement alone,
and
intersect at right angles, so, consequently,
.
If transversal crosses two parallel lines
and
, then same-side interior angles are supplementary, so
and
are supplementary angles. Also, corresponding angles are congruent, so
.
By Statement 1 alone, angles and
are congruent as well as supplementary; by Statement 2 alone,
and
are also supplementary as well as congruent. Two angles that are both supplementary and congruent are both right angles, so from either statement alone,
and
intersect at right angles, so, consequently,
.
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Find the equation of the line that is perpendicular to the following equation and passes through the point
.

Find the equation of the line that is perpendicular to the following equation and passes through the point .
Tap to reveal answer
To solve this equation, we want to begin by recalling how to find the slope of a perpendicular line. In this case, our original line is modeled by the following:

To find the slope of any line perpendicular to the above equation, we simply need to take the reciprocal of the first slope, and then change its sign. Our original slope is
, so

becomes
.
If we flip
, we get
, and the opposite sign of a negative is a positive; hence, our slope is positive
.
So, we know our perpendicular line should look something like this:

However, we need to find out what
(our
-intercept) is in order to complete our equation. To do so, we need to plug in the ordered pair we received in the question,
, and solve for
:




So, by putting everything together, we get our final equation:

This equation satisfies the conditions of being perpendicular to our initial equation and passing through
.
To solve this equation, we want to begin by recalling how to find the slope of a perpendicular line. In this case, our original line is modeled by the following:
To find the slope of any line perpendicular to the above equation, we simply need to take the reciprocal of the first slope, and then change its sign. Our original slope is , so
becomes
.
If we flip , we get
, and the opposite sign of a negative is a positive; hence, our slope is positive
.
So, we know our perpendicular line should look something like this:
However, we need to find out what (our
-intercept) is in order to complete our equation. To do so, we need to plug in the ordered pair we received in the question,
, and solve for
:
So, by putting everything together, we get our final equation:
This equation satisfies the conditions of being perpendicular to our initial equation and passing through .
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Which of the following lines is perpendicular to
?
Which of the following lines is perpendicular to ?
Tap to reveal answer
In order for a line
to be perpendicular to another line
defined by the equation
, the slope of line
must be a negative reciprocal of the slope of line
. Since line
's slope is
in the slope-intercept equation above, line
's slope would therefore be
.
In this instance,
, so
. Therefore, the correct solution is
.
In order for a line to be perpendicular to another line
defined by the equation
, the slope of line
must be a negative reciprocal of the slope of line
. Since line
's slope is
in the slope-intercept equation above, line
's slope would therefore be
.
In this instance, , so
. Therefore, the correct solution is
.
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What is the measure of an angle that is supplementary to a
angle?
What is the measure of an angle that is supplementary to a angle?
Tap to reveal answer
Supplementary angles have degree measures that total
, so an angle supplementary to
would measure
.
Supplementary angles have degree measures that total , so an angle supplementary to
would measure
.
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What is the measure of an angle that is complementary to a
angle?
What is the measure of an angle that is complementary to a angle?
Tap to reveal answer
Complementary angles have degree measures that total
, so an angle complementary to
would measure
.
Complementary angles have degree measures that total , so an angle complementary to
would measure
.
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What is the measure of an angle congruent to a
angle?
What is the measure of an angle congruent to a angle?
Tap to reveal answer
Congruent angles have degree measures that are equal, so an angle congruent to
is
.
Congruent angles have degree measures that are equal, so an angle congruent to is
.
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Consider segment
which passes through the points
and
.
Find the length of segment
.
Consider segment which passes through the points
and
.
Find the length of segment .
Tap to reveal answer
This question requires careful application of distance formula, which is really a modified form of Pythagorean theorem.

Plug in everthing and solve:

So our answer is 156.6
This question requires careful application of distance formula, which is really a modified form of Pythagorean theorem.
Plug in everthing and solve:
So our answer is 156.6
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What is the length of a line segment that starts at the point
and ends at the point
?
What is the length of a line segment that starts at the point and ends at the point
?
Tap to reveal answer
Using the distance formula for the length of a line between two points, we can plug in the given values and determine the length of the line segment by calculating the distance between the two points:





Using the distance formula for the length of a line between two points, we can plug in the given values and determine the length of the line segment by calculating the distance between the two points:
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What is the measure of an angle that is supplementary to a
angle?
What is the measure of an angle that is supplementary to a angle?
Tap to reveal answer
Supplementary angles have degree measures that total
. Since we have an
angle, the supplementary angle would measure 
Supplementary angles have degree measures that total . Since we have an
angle, the supplementary angle would measure
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Which of the following angles is complementary to an
angle?
Which of the following angles is complementary to an angle?
Tap to reveal answer
Complementary angles have degree measures that total
. Since we have an
angle, the supplementary angle would measure 
Complementary angles have degree measures that total . Since we have an
angle, the supplementary angle would measure
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Which of the following angles is congruent to a
angle?
Which of the following angles is congruent to a angle?
Tap to reveal answer
Congruent angles have the same degree measure, so an angle congruent to a
angle would also measure
.
Congruent angles have the same degree measure, so an angle congruent to a angle would also measure
.
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You are given Pentagon
such that:

and

Calculate 
You are given Pentagon such that:
and
Calculate
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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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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?
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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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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 .
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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 .
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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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