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pappueshwar
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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.
https://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
