Data Sufficiency

See attached picture. I understand after seeing the answer that the key is finding the 30-60-90 property of triangle (RPT)--through this you can determine angle (RSQ) and angle (UST) and subtract from 180 to find angle x. Answer C.

But how do you determine the 30-60-90 relationship of the largest triangle with both statement 1 and 2?

jswesth wrote:See attached picture. I understand after seeing the answer that the key is finding the 30-60-90 property of triangle (RPT)--through this you can determine angle (RSQ) and angle (UST) and subtract from 180 to find angle x. Answer C.

But how do you determine the 30-60-90 relationship of the largest triangle with both statement 1 and 2?

Hi
there is nothing much to do with 30-60-90 triangle. you can form various right triangle with different combinations
The KEY here is that both the statements combined tell you that the two triangles RQS and SUT formed are isosceles ..
Now it is all about finding angles..

1) RQS is isosceles because rq = rs, so angle rqs = rsq = say a..
so angle r = 180 - 2a..

2) similarly in SUT
SUT is isosceles because ut = ts, so angle tus = tsu = say b..
so angle t = 180 - 2b..

3) take the bigger triangle PRT
angle p +r+t=180..
substitute values from i and 2 above and p=90
90 + 180 -2a + 180 - 2b =180..
2(a+b) = 270 or a+b=135..

Finally look at x angle -
x+a+b = 180..
substitute a+b = 135
x+135 = 180
x =45

Hope it helps

Ah, of course. Thank you sir.