DSQ: Understanding intersecting lines - GMAT Quantitative Reasoning
Card 0 of 7
Refer to the above figure. Jane chose one of the line segments shown in the above diagram but she will not reveal which one. Which one did she choose?
Statement 1: One of the endpoints of the line segment is 
.
Statement 2: The line segment includes 
.
Refer to the above figure. Jane chose one of the line segments shown in the above diagram but she will not reveal which one. Which one did she choose?
Statement 1: One of the endpoints of the line segment is
Statement 2: The line segment includes
If we know both statements, then we know that the segment can be either 
or 
, since each has endpoint 
and each includes 
; we can not eliminate either, however.
If we know both statements, then we know that the segment can be either
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How many times does 
and 
intersect?
I) 
is a linear equation with a slope of 
.
II) 
is quadratic equation with a vertex at 
.
How many times does
I)
II)
When we have a linear equation and a quadratic equation there are only so many times they can intersect. They can intersect 0 times, once, or twice.
I) Gives us the slope of one equation.
II) Gives us the vertex of our quadratic equation.
If you draw a picture, it should be apparent that we don't have enough information to know exactly how many times they intersect. Our quadratic could be facing up or down, and our linear equation could go straight through both arms, or it could miss it entirely. Therfore, neither statement is sufficient.
When we have a linear equation and a quadratic equation there are only so many times they can intersect. They can intersect 0 times, once, or twice.
I) Gives us the slope of one equation.
II) Gives us the vertex of our quadratic equation.
If you draw a picture, it should be apparent that we don't have enough information to know exactly how many times they intersect. Our quadratic could be facing up or down, and our linear equation could go straight through both arms, or it could miss it entirely. Therfore, neither statement is sufficient.
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Find the 4 angles created by the two intersecting lines.
Statement 1: 
and 
Statement 2: 
and 
Find the 4 angles created by the two intersecting lines.
Statement 1:
Statement 2:
Statement 1: 
and 
The line 
is a horizontal line on the x-axis. The line 
is a vertical line graphed along the y-axis. The lines will create perpendicular angles, which are all 90 degrees.
Statement 2: 
and 
These two functions are in 
form, which allows us to determine the slopes of these functions. The slopes are 2 and negative half, which are both the negative reciprocal to each other. The property of negative reciprocal slopes state that these two lines are also perpendicular to each other.
Therefore:
Statement 1:
The line
Statement 2:
These two functions are in
Therefore:
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Determine the value of the four angles created by the intersecting lines.
Statement 1: Two angles are acute, and two angles are obtuse.
Statement 2: Any two non-perpendicular intersecting lines with known equations.
Determine the value of the four angles created by the intersecting lines.
Statement 1: Two angles are acute, and two angles are obtuse.
Statement 2: Any two non-perpendicular intersecting lines with known equations.
Statement 1): Two angles are acute, and two angles are obtuse.
This statement is not necessarily true. Two intersecting lines may also be perpendicular to each other, which means that all four angles are 90 degrees.
There is not enough information to justify this statement.
Statement 2: Any two non-perpendicular intersecting lines with known equations.
This is a tricky statement.
When two functions meet, they must have an intersecting point 
. Both functions 
can be set equal to each other to determine that intersecting point.
Draw an imaginary line 
where the line is perpendicular to the first function and passes through the second function at some known arbitrary point 
. Point 
will need to be determined.
The equation of the third function can be determined since imaginary line 
intersects equation 
at 
, and 
is also perpendicular to 
. The slope of can be determined since it's the negative reciprocal of the slope of .
After the equation 
has been determined by using point 
and the slope of 
, the point 
can also be determined by setting the functions 
equal to each other.
Once the points 
have been determined, the distance formula may be used to determine the lengths from 
, 
, and 
.
The Law of Sines can then be used to determine the interior angles of the triangle bounded by 
. Knowing one angle at the intersection of 
is sufficient to solve for all four angles by supplementary and opposite angle rules.
Therefore:
Statement 1): Two angles are acute, and two angles are obtuse.
This statement is not necessarily true. Two intersecting lines may also be perpendicular to each other, which means that all four angles are 90 degrees.
There is not enough information to justify this statement.
Statement 2: Any two non-perpendicular intersecting lines with known equations.
This is a tricky statement.
When two functions meet, they must have an intersecting point
Draw an imaginary line
The equation of the third function can be determined since imaginary line
After the equation
Once the points
The Law of Sines can then be used to determine the interior angles of the triangle bounded by
Therefore:
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What is the value of x?
(1) 
(2) 
What is the value of x?
(1)
(2)
With just statement 1, we know 
, so we could determine the measure of angle y, but there is no definitive relationship between z and x nor y and x, because we don't know if lines p and q are parallel.
With just statement 2,we know 
, so we could determine the measure of angle z, but there is no definitive relationship between y and x nor z and x, because we don't know if lines p and q are parallel.
Even if we have the information from both statements 1 and 2, we still do not know if lines p and q are parallel, therefore there is no difinitive relationship between angle y and angle x nor angle z and angle x.
With just statement 1, we know
With just statement 2,we know
Even if we have the information from both statements 1 and 2, we still do not know if lines p and q are parallel, therefore there is no difinitive relationship between angle y and angle x nor angle z and angle x.
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What is the value of x?
(1) 
(2) Lines p and q are parallel
What is the value of x?
(1)
(2) Lines p and q are parallel
With just statement 1, there is no definitive relationship between angle y and angle x.
With just statement 2, there is a difinitive relationship between angle y and angle x, but we don't know the measure of angle y.
If you have the information from both statements 1 and 2, you can determine the measure of 
, so 
.
Fortunately, this is a data sufficiency question, so you don't have to actually do the math, you just have to know that you have all the information to do the math.
With just statement 1, there is no definitive relationship between angle y and angle x.
With just statement 2, there is a difinitive relationship between angle y and angle x, but we don't know the measure of angle y.
If you have the information from both statements 1 and 2, you can determine the measure of
Fortunately, this is a data sufficiency question, so you don't have to actually do the math, you just have to know that you have all the information to do the math.
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Lines p and q are parallel. What's the value of x?
(1) 
(2) 
Lines p and q are parallel. What's the value of x?
(1)
(2)
Although the question itself tells us that lines p and q are parallel, the iinformation in statement 1 is insufficient to determine a definitive value for either y or z.
With the information from the question that lines p and q are parallel, and the added information from statement 2 that 
, using the rules of supplementary angles and alternate interior angles, we can determine the value of x.
Although the question itself tells us that lines p and q are parallel, the iinformation in statement 1 is insufficient to determine a definitive value for either y or z.
With the information from the question that lines p and q are parallel, and the added information from statement 2 that
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