GMAT Math : Understanding intersecting lines

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #1 : Understanding Intersecting Lines

Lines

Which is another name for  in the above diagram? 

Possible Answers:

Correct answer:

Explanation:

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.

Example Question #2 : Understanding Intersecting Lines

Lines

Which of the following is not a valid name for the triangle in the above diagram?

Possible Answers:

All of the names in the other four choices are valid names for the triangle.

Correct answer:

All of the names in the other four choices are valid names for the triangle.

Explanation:

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.

Example Question #2 : Understanding Intersecting Lines

Lines

Refer to the above diagram.

Which of the following is not a valid alternative name for  ?

Possible Answers:

Correct answer:

Explanation:

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.

Example Question #1 : Understanding Intersecting Lines

Lines

 and which angle form a linear pair?

Possible Answers:

There is no angle that forms a linear pair with .

Correct answer:

Explanation:

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.

Example Question #2 : Understanding Intersecting Lines

Lines

Note: Figure NOT drawn to scale.

Refer to the above diagram. Which is a valid alternative name for  ?

Possible Answers:

Correct answer:

Explanation:

A line segment is named after its two endpoints in either order;  is the segment with endpoints  and , so it can also be named 

Example Question #5 : Understanding Intersecting Lines

Lines

 and which angle are examples of a pair of vertical angles?

Possible Answers:

There is no angle that forms a vertical pair with .

Correct answer:

Explanation:

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

Example Question #6 : Understanding Intersecting Lines

At what point do  and  intersect?

Possible Answers:

Correct answer:

Explanation:

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 .

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