The answer is C - but can someone please explain why? Does the answer have something to do with similar triangles?

This is from GMATPrep (TM) Test.

Thanks.

File is attached.

P.S: It is difficult to read the explanation below without having the figure in front of you. Hence recommending to draw the figure and then read the explanation

First of all we know from the figure RPT is a right triange. So Angle RPT = 90, Let us assume angle PRT = Y (which is also angle QRS) and then angle RTP (or angle STU) = 90 - Y.

Statement (1): QR = RS means the triangle RQS is an isosceles triangle. So Angle RSQ and RQS are equal. In triange RQS we have already assumed angle QRS = Y. So the remaining two angles will sum to 180-Y. Hence angle RSQ = 90 - Y/2 and RQS = 90 - Y/2. Even now, we don't have any clue with regards to x. So this is not sufficient.

Statement (2): ST = TU means, triange STU is an isosceles triangle with angle TSU = angle TUS. We already assumed angle STU = 90 - Y. So the remaining angles will sum to 180 - (90-Y) which is 90 + Y. So angle TSU and TUS will be each 45 + Y/2.

Still we don't have any clue with regards to x, so this is not sufficient.

Now combining (1) and (2). If you look at the figure,

Angle RSQ + Angle QSU + Angle SUT = 180 degrees

That is (90 - Y/2) + x + (45 + Y/2) = 180.

The Y/2 will cancel out.

135 + x = 180
Hence x = 180 - 135 = 45 degrees.

So we are able to find x. Answer is (C).

I haven't used the properties of similar triangles, and honestly I couldn't find one. But I was able to solve it by using following two properties: -

1) Sum of the four angles of quadrilateral = 360
2) Sum of angles made by intersecting line or lines on a straight line = 180

Here is the solution

In Quadrilateral PQSU (Property 1) PQS + x + SUP + 90 = 360--------(1)

Also angle (Property 2) PQS = 180 - RQS and angle SUP = 180 - SUT

Putting values in (1) 180 - RQS + x + 180 - SUT + 90 = 360
So , x + 90 - RQS - SUT = 0----(2)

We also know angle RSQ + x + TSU = 180-----(3)

From statement (1) we know Angle RSQ = Angle RQS
From statement (2) we know Angle SUT = Angle TSU

Adding (2) & (3) and from information from statements (1) & (2)

2x + 90 = 180, x can be calculated

Hence (C) is the answer

Hope it helps

GMAT 2007

Two methods of approaching the same problem is presented, which one of the two is preferable to use on a timed exam?

Thanks

i knew we had this sitting around here somewhere

booya!

whichever one you think of first.

this is serious - no sarcasm intended. the gmat is famous for including problems that can be solved by an impressive variety of methods, so it's likely that whatever method you may happen upon first will work. you may not think of the most elegant solution right away, but, fortunately, there are no points for style.

whatever solution you find, the important part is this: do not give up in the middle of a solution, unless you are ABSOLUTELY sure that the solution is a dead end.