DS Geometry

[prachi18oct]

In the quadrilateral above, what is the measure of angle x?

(1) y = 75
(2) The quadrilateral MNQR is a reflection of quadrilateral QOPR around line QR.



Shouldn't the OA be
C ?

[bubbliiiiiiii]

B?

Since its a reflection, we can say that quadrilateral is a trapezium with NO parallel to MP. Now, its possible to get X.

[GMATGuruNY]

In the quadrilateral above, what is the measure of angle x?

(1) y = 75
(2) The quadrilateral MNQR is a reflection of quadrilateral QOPR around line QR.

Statement 2:
When a figure is reflected about a line, the result is SYMMETRY about the line.
Thus, the following figure is implied:

As shown in the figure above, x=75.
SUFFICIENT.

Statement 1:
The figure could look like this:

In this case, x=75.

The figure could look like this:

In this case, x=90.

Since x can be different values, INSUFFICIENT.

The correct answer is B.

[prachi18oct]

In the quadrilateral above, what is the measure of angle x?

(1) y = 75
(2) The quadrilateral MNQR is a reflection of quadrilateral QOPR around line QR.

Statement 2:
When a figure is reflected about a line, the result is SYMMETRY about the line.
Thus, the following figure is implied:

As shown in the figure above, x=75.
SUFFICIENT.

Statement 1:
The figure could look like this:

In this case, x=75.

The figure could look like this:

In this case, x=90.

Since x can be different values, INSUFFICIENT.

The correct answer is B.

Hi GMATGuruNY,

Is it necessary that the QR be perpedicular to NO and MP? OG says "Lines shown as straight can be assumed to be straight" so we can safely assume that QR and NO are perpendicular and so are Qr and MP ?

[GMATGuruNY]

Hi GMATGuruNY,

Is it necessary that the QR be perpedicular to NO and MP? OG says "Lines shown as straight can be assumed to be straight" so we can safely assume that QR and NO are perpendicular and so are Qr and MP?

We can safely assume that NO, OP, MP, NM and QR are straight lines.
We CANNOT assume that any of these straight lines are parallel or of equal length.
The third figure in my post above illustrates a case in which NO is not parallel to MP:


Statement 2, however, requires that NO be parallel to MP.
Proof:
Since statement 2 implies symmetry about QR, ∠NQR = ∠OQR.
Since ∠NQR + ∠OQR = 180, ∠NQR = ∠OQR = 90.
The same line of reasoning can be used to show that ∠MRQ = ∠PRQ = 90.
The following figure is yielded:

Since NO and MP are both perpendicular to the same line -- QR -- NO and MP must be PARALLEL.

[OptimusPrep]

In the quadrilateral above, what is the measure of angle x?

(1) y = 75
(2) The quadrilateral MNQR is a reflection of quadrilateral QOPR around line QR.

Let us take the Statement 1:
It just gives us y = 75. But we are not sure if this is a trapezoid (although it may look like, but no where it is mentioned that PM is || to ON) or not, hence we cannot conclude anything for the angle x

Statement2:
The quadrilateral MNQR is a reflection of quadrilateral QOPR around line QR.
This means angle RMP = angle RPO = 105
And angle MNQ = angle POQ

We also know that the sum of 4 angles of a quadrilateral = 360.
So 105 + 105 + x + y = 360
=> x+x = 150
Hence x = 75.

So, the correct option is B

[nikhilgmat31]

Not able to see diagram

[Satya.Achanta]

I though, since in 1st option given y=75, x would also be 75 because they look like same in the diagram.and the lines MN and PO are parallel. Can you let me know how to interpret these kinda problems ?, here why cant't be y=x ?

[GMATGuruNY]

I though, since in 1st option given y=75, x would also be 75 because they look like same in the diagram.and the lines MN and PO are parallel. Can you let me know how to interpret these kinda problems ?, here why cant't be y=x ?

Figures for PS problems are generally drawn to scale.
The same CANNOT be said for DS figures, which are often NOT drawn to scale.
In many cases, figures in DS problems are provided not to help us but to FOOL us.
Here, the provided figure LOOKS symmetrical about QR in order to FOOL us into assuming that x=y.
But -- illustrated in my post above -- the figure in statement 1 could look like this:

In this case, x≠y.

When solving DS geometry problems:
Trust only what you read, not what you see.
Here, nothing stated in the prompt or in statement 1 requires that x=y.

[Max@Math Revolution]

In DS, Variable approach is the easitest way to find the answer without actually solving the problem.

Remember equal number of variables and equations ensures a solution.

In original condition, angle N + angle M + angle O + 105 =360.
We have 3 variable (N,M,O) and 1 equation. we need 2 more equations. So C is the answer.

Why C? If you know our own unique logic, you don't need to actually solve it. But let me prove this for you
In (1)&(2), x=y=75 it is sufficient so C is the answer.

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[Max@Math Revolution]

In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem.
Remember equal number of variables and equations ensures a solution.


In original condition, angle N+angle M+angle O+105=360. We have 3 variable and 1 equation we need 2 more equations. So C is the answer.


Why C? If you know our own innovative logics to find the answer, you don't need to actually solve the problem.

www.mathrevolution.com

Regardless of your math skills, our world's first Variable Approach (DS) and IVY Approach (PS) help students dramatically reduce their time spent per question and improve accuracy. You will have 10 min. to spare before the test ends.

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Hi Max@MathRevolution,

This DS prompt does NOT require a 'system' of equations to answer the given question. Given the geometry rules involved, Fact 2 actually provides enough information on its own, so the answer is NOT C.

GMAT assassins aren't born, they're made,
Rich