What is the value of x + y in the figure above?

What is the value of x + y in the figure above?

Sum of Interior Angles of a polygon is \(180(n-2)\) where \(n\) is the number of sides (so is the number of angles). So, the sum of the interior angles of a quadrilateral is 180*2=360. Look at the diagram below:



According to the above: (180-x)+(180-y)+(180-z)+(180-w)=360 --> x+y=360-w-z, so all we need to know is the values of w and z.

(1) w= 95. Not sufficient.
(2) z = 125. Not sufficient.

(1)+(2) Sufficient.

Answer: C.


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In line to Bunuel's solution, we can also solve it using exterior angle and vertical angle properties. The sum of all exterior angles for a polygon is 360. So x+y = 360 - (w+z), and both 1 and 2 together provide the same.

Target question: What is the value of x + y?

Statement 1: w = 95

Important: For geometry DS questions, we are typically checking to see whether the statements "lock" a particular angle or length into having just one value. This concept is discussed in much greater detail in the second video below.

If w = 95, then the angle inside the quadrilateral must be 85.

So, those 2 angles (95 and 85) are "locked." In other words, the 2 lines that create those two angles are locked in place to create the 95- and 85-degree angles.
However, line1 is not locked into place, so we can still move it, which means we can freely alter the size of angle y.
As such, the value of x + y will vary.
Since we cannot answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: z = 125
If z = 125, then the angle inside the quadrilateral must be 55.

Since line2 is not locked into place, we can still move it, which means we can freely alter the size of angle x.
As such, the value of x + y will vary.
Since we cannot answer the target question with certainty, statement 2 is SUFFICIENT

Statements 1 and 2 combined:
We now have the following:

Since all angles in a quadrilateral add to 360 degrees, we know that 85 + 55 + j + k = 360
If we solve for j + k, we get: j + k = 220

Also notice that, since angles x and k are on a line, it must be true that x + k = 180.
Similarly, it must be true that y + j = 180
If we combine both of these equations, we get: x + y + j + k = 360
Since we already know that j + k = 220, we can replace j + k with 220, to get:
x + y + 220 = 360
This means x + y = 140
Since we cannot answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

RELATED VIDEOS



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If (180-w) = a and (180-z) = b, then the below generalized form is always true for any quadrilateral:
a+b=x+y

Maybe we can call it exterior angle theorem for quadrilaterals.

How did you get from (180-x)+(180-y)+(180-z)+(180-w)=360 to x+y=360-w-z?

I am getting the following:

(180-z)+(180-w)=-180+x-180+y+360
-z-w=x+y+360
x+y=-360-z-w

Can you show more detail? Thanks in advance

\((180-x)+(180-y)+(180-z)+(180-w)=360\)

\(720 - x - y - z - w = 360\)

\(360 - x - y - z - w=0\)

\(360-w-z=x+y\)
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Similar: https://gmatclub.com/forum/in-the-figur ... 40190.html

Solution:

We need to determine the value of x + y. We see that x and y are two of the exterior angles of the quadrilateral, and w and z are the other two exterior angles. Notice that x + y + w + z = 360 since the sum of the measures of the exterior angles of a polygon is always 360 degrees. Therefore, if we know the values of both w and z or if we know the value of w + z, then we can determine the value of x + y.

Statement One Alone:

Since we don’t know the value of z, statement one alone is not sufficient.


Statement Two Alone:

Since we don’t know the value of w, statement two alone is not sufficient.

Statements One and Two Together:

Recall that that x + y + w + z = 360; therefore, we have:

x + y + 95 + 125 = 360

x + y + 220 = 360

x + y = 140

Both statements together are sufficient.

Answer: C
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Question - if the y angle measure was actually placed on the top right corner of the polygon, answer would be E, right?

Yup, you'd be able to find the value of (x-y) in that case and not (x+y).

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