As is often the case with difficult triangle questions, solving this one requires reliance on the sum of angles rule. Specifically, the sum of angles in PQSU is 360

To solve you must also realize that in any given triangle, if two sides are equal, the angles that face the sides are also equal. So (1) tells us that in QRS, angles Q & S are equal. (2) tells us that in STU, angles S & U are equal. Together, these provide a way to solve if you realize that the sum of the three angles at point S is 180 (straight line). C is correct.

It's tough to illustrate in words; please review the video explanation: this is GMATPrep question 1080. If you have access to the Solutions Engine, you can find and practice similar questions by searching for questions of topic="geometry" and difficult="700+"

Hope this helps,
-Patrick

I think the answer is D

We can solve for the angle by following equations:

say <RQS = <RSQ = y
<SUP = z

PQSU = 360
(180-y)+x+(90+z) = 360 --> x+z-y =90 -----1

Ext angle = sum of int opp angles
z = 180-x-y + 90-2y ---> x+3y+z=270 ----- 2

QSTP =360
90 + 90-2y + 180-y + 180-y = 360 ----> 180/4 = y ---------3

Hence,
x can be solved

Similarily for the second statement we can solve for x,

Hence D is the answer

Maqavi I like your approach but you've made mistakes. The official answer was provided in the original post. It's not D. I can see 2 errors in your post:

1) You found the measure of angle T as 90-2y. This is not correct. T + P + R should be 180. Since R is 180-2y and P is 90, T must be 2y-90 instead of what you have.

2) In your last line (equation 3), it looks like you replaced x with 180-y. This is not correct.

Please have a look at the written/video solution from my previous post for detailed explanation.

-Patrick

I agree with Patrick !! i m deleting my solution ...

I liked your approach though maqavi; to try to figure out as many of the angles as possible by incorporating the info you're given.

Statement 1:

Ang RQS = Ang RSQ
This tells us nothing about x and Angles SUP and QST - INSUFFICIENT

Statement 2:

Ang TSU = Ang TUS
This tells us nothing about x and Angles SQP and RSQ - INSUFFICIENT


Both together:

Let angle RQS = Angle RSQ = y
SQP = 180 - y
Given QPU = 90
SUP = 360 - (SQP + QPU + x)
= 360 - (180 - y + 90 + x)
= 90 + y - x

SUP + SUT = 180
=> SUT = 180 - (90+y-x)
= 90 - y + x

UST = SUT = 90 - y + x

UST + x + RSQ = 180
90 - y + x + x + y = 180
2x = 90
SUFFICIENT.

albatross86 wrote:Statement 1:

Ang RQS = Ang RSQ
This tells us nothing about x and Angles SUP and QST - INSUFFICIENT

Statement 2:

Ang TSU = Ang TUS
This tells us nothing about x and Angles SQP and RSQ - INSUFFICIENT


Both together:

Let angle RQS = Angle RSQ = y
SQP = 180 - y
Given QPU = 90
SUP = 360 - (SQP + QPU + x)
= 360 - (180 - y + 90 + x)
= 90 + y - x

SUP + SUT = 180
=> SUT = 180 - (90+y-x)
= 90 - y + x

UST = SUT = 90 - y + x

UST + x + RSQ = 180
90 - y + x + x + y = 180
2x = 90
SUFFICIENT.

i agree with your line of thinking and your answer. this is how got the same result:

RSQ+x = SUT + T
UST+x = RQS + R

adding above equations:

RSQ+x+UST+x = SUT+RQS + T+R
RSQ+UST+2x = SUT+RQS+90
2x = 90

ANSWER (C)