Understanding intersecting lines - GMAT Quantitative
Card 1 of 56
Which is another name for 
in the above diagram?
Which is another name for
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The middle letter of an angle must be its vertex, so the vertex of 
is 
.
An angle can be named after its vertex alone as long as it is the only angle with that vertex. This is not the case here, so 
cannot be correct. Also, 
and 
are incorrect since the middle letters are not 
.
The first and last letters of a three-letter angle name must be points on different sides of the angle. 
is incorrect since 
and 
are on the same side of the angle. The correct choice is therefore 
, since 
and 
are on different sides of the angle.
The middle letter of an angle must be its vertex, so the vertex of
An angle can be named after its vertex alone as long as it is the only angle with that vertex. This is not the case here, so
The first and last letters of a three-letter angle name must be points on different sides of the angle.
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and which angle are examples of a pair of vertical angles?
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Two angles are vertical if they have the same vertex and if their sides form two pairs of opposite rays. The correct choice will have vertex 
, which is the vertex of 
. Its rays will be the rays opposite 
and 
, which, are, respectively, 
and 
, respectively. The angle that fits this description is 
.
Two angles are vertical if they have the same vertex and if their sides form two pairs of opposite rays. The correct choice will have vertex
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Which of the following is not a valid name for the triangle in the above diagram?
Which of the following is not a valid name for the triangle in the above diagram?
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A triangle can be named after its three vertices in any order. Since the vertices of the triangle are 
, any name that includes these three letters is valid. All of the choices fit this description.
A triangle can be named after its three vertices in any order. Since the vertices of the triangle are
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Refer to the above diagram.
Which of the following is not a valid alternative name for 
?
Refer to the above diagram.
Which of the following is not a valid alternative name for
Tap to reveal answer
An angle can be given a name with three letters if the middle letter is the name of the vertex and the other two letters denote points on different sides of the angle. All four of the three-letter choices fit this description.
An angle can be given a one-letter name if the letter is the name of the vertex and if it is the only angle shown in the diagram to have that vertex (thereby avoiding ambiguity). There are four angles in the diagram with vertex 
, so we cannot use 
to indicate any of them, including 
. 
is the correct choice.
An angle can be given a name with three letters if the middle letter is the name of the vertex and the other two letters denote points on different sides of the angle. All four of the three-letter choices fit this description.
An angle can be given a one-letter name if the letter is the name of the vertex and if it is the only angle shown in the diagram to have that vertex (thereby avoiding ambiguity). There are four angles in the diagram with vertex
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and which angle form a linear pair?
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Two angles form a linear pair if they have the same vertex, they share one side, and their interiors do not intersect. 
has vertex 
and sides 
and 
; 
has vertex 
, shares side 
, and shares no interior points, so this is the correct choice.
Two angles form a linear pair if they have the same vertex, they share one side, and their interiors do not intersect.
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Note: Figure NOT drawn to scale.
Refer to the above diagram. Which is a valid alternative name for 
?
Note: Figure NOT drawn to scale.
Refer to the above diagram. Which is a valid alternative name for
Tap to reveal answer
A line segment is named after its two endpoints in either order; 
is the segment with endpoints 
and 
, so it can also be named 
.
A line segment is named after its two endpoints in either order;
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At what point do 
and 
intersect?
At what point do
Tap to reveal answer
To find where two lines intersect, simply set them equal to each other and solve for 
. Then plug the resulting 
value back in to one of the equations and solve for 
.
Add 
to both sides and subtract 
from both sides to isolate our like terms:
So, 
must be true for where these lines intersect. Next, plug 
back in for 
in one of our original equations:
So, the 
value of our intersection is 
.
This makes the coordinate of our intersection 
.
You can check your answer by plugging in the point you calculated into both equations. Both equations will be true when 
is equal to 
and 
—in this case, 
and 
—is equal to 
.
To find where two lines intersect, simply set them equal to each other and solve for
Add
So,
So, the
This makes the coordinate of our intersection
You can check your answer by plugging in the point you calculated into both equations. Both equations will be true when
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Which is another name for in the above diagram?
Which is another name for
Tap to reveal answer
The middle letter of an angle must be its vertex, so the vertex of is .
An angle can be named after its vertex alone as long as it is the only angle with that vertex. This is not the case here, so cannot be correct. Also, and are incorrect since the middle letters are not .
The first and last letters of a three-letter angle name must be points on different sides of the angle. is incorrect since and are on the same side of the angle. The correct choice is therefore , since and are on different sides of the angle.
The middle letter of an angle must be its vertex, so the vertex of
An angle can be named after its vertex alone as long as it is the only angle with that vertex. This is not the case here, so
The first and last letters of a three-letter angle name must be points on different sides of the angle.
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and which angle are examples of a pair of vertical angles?
Tap to reveal answer
Two angles are vertical if they have the same vertex and if their sides form two pairs of opposite rays. The correct choice will have vertex , which is the vertex of . Its rays will be the rays opposite and , which, are, respectively, and , respectively. The angle that fits this description is .
Two angles are vertical if they have the same vertex and if their sides form two pairs of opposite rays. The correct choice will have vertex
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Which of the following is not a valid name for the triangle in the above diagram?
Which of the following is not a valid name for the triangle in the above diagram?
Tap to reveal answer
A triangle can be named after its three vertices in any order. Since the vertices of the triangle are , any name that includes these three letters is valid. All of the choices fit this description.
A triangle can be named after its three vertices in any order. Since the vertices of the triangle are
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Refer to the above diagram.
Which of the following is not a valid alternative name for ?
Refer to the above diagram.
Which of the following is not a valid alternative name for
Tap to reveal answer
An angle can be given a name with three letters if the middle letter is the name of the vertex and the other two letters denote points on different sides of the angle. All four of the three-letter choices fit this description.
An angle can be given a one-letter name if the letter is the name of the vertex and if it is the only angle shown in the diagram to have that vertex (thereby avoiding ambiguity). There are four angles in the diagram with vertex , so we cannot use to indicate any of them, including . is the correct choice.
An angle can be given a name with three letters if the middle letter is the name of the vertex and the other two letters denote points on different sides of the angle. All four of the three-letter choices fit this description.
An angle can be given a one-letter name if the letter is the name of the vertex and if it is the only angle shown in the diagram to have that vertex (thereby avoiding ambiguity). There are four angles in the diagram with vertex
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and which angle form a linear pair?
Tap to reveal answer
Two angles form a linear pair if they have the same vertex, they share one side, and their interiors do not intersect. has vertex and sides and ; has vertex , shares side , and shares no interior points, so this is the correct choice.
Two angles form a linear pair if they have the same vertex, they share one side, and their interiors do not intersect.
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Note: Figure NOT drawn to scale.
Refer to the above diagram. Which is a valid alternative name for ?
Note: Figure NOT drawn to scale.
Refer to the above diagram. Which is a valid alternative name for
Tap to reveal answer
A line segment is named after its two endpoints in either order; is the segment with endpoints and , so it can also be named .
A line segment is named after its two endpoints in either order;
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At what point do and intersect?
At what point do
Tap to reveal answer
To find where two lines intersect, simply set them equal to each other and solve for . Then plug the resulting value back in to one of the equations and solve for .
Add to both sides and subtract from both sides to isolate our like terms:
So, must be true for where these lines intersect. Next, plug back in for in one of our original equations:
So, the value of our intersection is .
This makes the coordinate of our intersection .
You can check your answer by plugging in the point you calculated into both equations. Both equations will be true when is equal to and —in this case, and —is equal to .
To find where two lines intersect, simply set them equal to each other and solve for
Add
So,
So, the
This makes the coordinate of our intersection
You can check your answer by plugging in the point you calculated into both equations. Both equations will be true when
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Which is another name for in the above diagram?
Which is another name for
Tap to reveal answer
The middle letter of an angle must be its vertex, so the vertex of is .
An angle can be named after its vertex alone as long as it is the only angle with that vertex. This is not the case here, so cannot be correct. Also, and are incorrect since the middle letters are not .
The first and last letters of a three-letter angle name must be points on different sides of the angle. is incorrect since and are on the same side of the angle. The correct choice is therefore , since and are on different sides of the angle.
The middle letter of an angle must be its vertex, so the vertex of
An angle can be named after its vertex alone as long as it is the only angle with that vertex. This is not the case here, so
The first and last letters of a three-letter angle name must be points on different sides of the angle.
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and which angle are examples of a pair of vertical angles?
Tap to reveal answer
Two angles are vertical if they have the same vertex and if their sides form two pairs of opposite rays. The correct choice will have vertex , which is the vertex of . Its rays will be the rays opposite and , which, are, respectively, and , respectively. The angle that fits this description is .
Two angles are vertical if they have the same vertex and if their sides form two pairs of opposite rays. The correct choice will have vertex
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Which of the following is not a valid name for the triangle in the above diagram?
Which of the following is not a valid name for the triangle in the above diagram?
Tap to reveal answer
A triangle can be named after its three vertices in any order. Since the vertices of the triangle are , any name that includes these three letters is valid. All of the choices fit this description.
A triangle can be named after its three vertices in any order. Since the vertices of the triangle are
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Refer to the above diagram.
Which of the following is not a valid alternative name for ?
Refer to the above diagram.
Which of the following is not a valid alternative name for
Tap to reveal answer
An angle can be given a name with three letters if the middle letter is the name of the vertex and the other two letters denote points on different sides of the angle. All four of the three-letter choices fit this description.
An angle can be given a one-letter name if the letter is the name of the vertex and if it is the only angle shown in the diagram to have that vertex (thereby avoiding ambiguity). There are four angles in the diagram with vertex , so we cannot use to indicate any of them, including . is the correct choice.
An angle can be given a name with three letters if the middle letter is the name of the vertex and the other two letters denote points on different sides of the angle. All four of the three-letter choices fit this description.
An angle can be given a one-letter name if the letter is the name of the vertex and if it is the only angle shown in the diagram to have that vertex (thereby avoiding ambiguity). There are four angles in the diagram with vertex
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and which angle form a linear pair?
Tap to reveal answer
Two angles form a linear pair if they have the same vertex, they share one side, and their interiors do not intersect. has vertex and sides and ; has vertex , shares side , and shares no interior points, so this is the correct choice.
Two angles form a linear pair if they have the same vertex, they share one side, and their interiors do not intersect.
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Note: Figure NOT drawn to scale.
Refer to the above diagram. Which is a valid alternative name for ?
Note: Figure NOT drawn to scale.
Refer to the above diagram. Which is a valid alternative name for
Tap to reveal answer
A line segment is named after its two endpoints in either order; is the segment with endpoints and , so it can also be named .
A line segment is named after its two endpoints in either order;
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