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Hi, there. I'm happy to help with this.

This is a trick geometry problem. Let angle R = r, and angle T = t. Notice that r + t = 90.

Statement #1: The length of line segment of QR is equal to the length of line segment RS

This statement implies that angle RQS = angle RSQ. Since angle R = r, we know angle RQS = angle RSQ = (180 - r)/2 = 90 - r/2. We know something about angle RSQ, but not about angle TSU. Therefore, we can't figure out anything about angle QSU. Statement #1, by itself, is insufficient.


Statement #2: The length of line segment of ST is equal to the length of line segment TU

Similar analysis to that of Statement #1. This statement implies that angle TSU = angle TUS. Since angle T = t, we know angle TSU = angle TUS = (180 - t)/2 = 90 - t/2. We know something about angle TSU, but not about angle RSQ. Therefore, we can't figure out anything about angle QSU. Statement #2, by itself, is insufficient.

Combined Statements #1 & #2:
Now, from the above analyses, we know
angle RSQ = 90 - r/2
angle TUS = 90 - t/2

We know that
(angle RSQ) + (angle QSU) + (angle TUS) = 180
(90 - r/2) + x + (90 - t/2) = 180
180 - (r + s)/2 + x = 180
- (r + s)/2 + x = 0
x = (r + s)/2
Now, remember from above, (r + s) = 90. Therefore:
x = 90/2 = 45

Therefore, with the two statements combined, we are able to determined the exact value of x.

Answer = C

Does that make sense?

Here's another DS questions that draws on the Isosceles Triangle Theorem.

http://gmat.magoosh.com/questions/1013

When you submit your answer to that question, the following page will have a full video explanation.

Please let me know if you have any questions on what I've said.

Mike

_________________
Magoosh GMAT Instructor
http://gmat.magoosh.com/

Note that the measure of angle x depends upon the position of points Q, S and U only. Unless we don't know the fixed positions of these three points, we cannot uniquely determine the measure of angle x.

Statement 1: QR = RS
Thus position of Q and S is fixed. But U can be any point on PT and accordingly value of x will be different; NOT sufficient.

Statement 2: ST = TU
Thus position of S and U is fixed. But Q can be any point on PR and accordingly value of x will be different; NOT sufficient.

1 & 2 Together: Now the three points are fixed. Let's see whether we can find x. Refer to the image below.



On point S, the sum of the three angles must be equal to 180°.
Thus, (x + y + z) = 180° ..................................... (i)

angle PQS = (180° - angle RQS) = (180° - z)
angle PUS = (180° - angle TUS) = (180° - y)

Now in quadrilateral PQSU,

Sum of all the internal angles = 360°
=> [x + 90° + (180° - y) + (180° - z)] = 360°
=> (x - y - z + 90°) = 0 .................................. (ii)

Now add (i) and (ii) => (2x + 90°) = 180° => x = 45° ; SUFFICIENT.

The correct answer is C.

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Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

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