OK folks, I'm sure this has been done before but can someone

Question

OK folks, I'm sure this has been done before but can someone shed some light on why the answer is what it is?

misterJJ2u · Answer

(bump) Anyone? Anyone? Bueller?

UMB · Answer

It is C.

If Angle R=2a, then angle RSQ = 90-a

If Angle T=2b, then angle TSU=90-b

For the main Triangle PRT: 90+2a+2b=180
so 2(a+b)=90 and a+b=45

<RSQ+X+<TSU=180 from here (90-a)+X+(90-b)=180

 X=a+b=45

misterJJ2u · Answer

Thanks UMB. 

Your reasoning cleared the fog for me but I didn't exactly follow the same path as you...I didn't understand your concept for R=2a. Shouldn't R=180-2a?

From the statements:
 (1) QR = RS, so angle RQS = angle RSQ; Angle QRS is unknown. however, angle QRS + angle RQS + angle QSR = 180. Since Q and S are the same, then QRS = 180-2a, where angle a = angle RQS = angle RSQ.  
 (2) ST = TU, so angle SUT = angle TSU; Angle STU is unknown. however, angle UTS + angle TSU + angle SUT = 180. Since S and U are the same, then UTS = 180-2b  

TOGETHER: 
Box inside the triangle:
360 = 90 + (180-a) + x + (180-b) which gives a + b - x = 90

From line segment RT w/ intersecting line segment QS:
180 = a + x + b which is the same as: a + b + x = 180

With these 2 equations, a + b can be cancelled out, leaving us with 2x = 90. Therefore x = 45 and thus why the answer is C.

ian7777 · Answer

C

I think mine is basically the same, but I didn't need to solve for x. A bit of a different approach. See attached. 

By the way, I'm taking it as a given that the statements alone are not enough.

misterJJ2u · Answer

Thanks Ian. that's the  simpler  version of my explanation...and simpler is better!!!

OK folks, I'm sure this has been done before but can someone

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