GMAT Practice Test 1 - DS Geometry Problem

[nicandalf]

I'm having trouble with the below problem even though I picked the right answer. I guess I'm forgetting my basic geometry rules.
Can someone please go through the steps and explain the problem?

[DanaJ]

Each stmt by itself is not sufficient, but if you put the two stmts together you get that you have 2 isosceles triangles: QRS and STU. This is why you will get two equations:
a. RSQ = (180 - QRS)/2, since RQS = RSQ
b. UST = (180 - STU)/2, since UST = SUT
Now, remember that you've got a right triangle, with RPT a right angle. This means that PRT + PTR = 180 - RPT = 90. But PRT is the same as QRS and PTR is the same as STU. You must also notice that:

x = 180 - RSQ - UST

Replace RSQ and UST and you get that:

x = 180 - (180 - PRT)/2 - (180 - PTR)/2 = 180 - [360 - (PRT + PTR)]/2 = 180 - (360 - 90)/2 = 180 - 270/2 = 180 - 135 = 45.

[nicandalf]

DanaJ, thank you so much, your explanation was perfect. I was a little unsure with my approach, but I can see the solution clearly now.

[mathlete1]

Each stmt by itself is not sufficient, but if you put the two stmts together you get that you have 2 isosceles triangles: QRS and STU. This is why you will get two equations:
a. RSQ = (180 - QRS)/2, since RQS = RSQ
b. UST = (180 - STU)/2, since UST = SUT
Now, remember that you've got a right triangle, with RPT a right angle. This means that PRT + PTR = 180 - RPT = 90. But PRT is the same as QRS and PTR is the same as STU. You must also notice that:

x = 180 - RSQ - UST

Replace RSQ and UST and you get that:

x = 180 - (180 - PRT)/2 - (180 - PTR)/2 = 180 - [360 - (PRT + PTR)]/2 = 180 - (360 - 90)/2 = 180 - 270/2 = 180 - 135 = 45.

DanaJ,

I'm having trouble with this one still. Why do you make a distinction between RSQ and QRS? or between UST and STU in the first part? Furthermore, why are you dividing by 2? Can you explain why you do this? Thank you

[GMATGuruNY]

I posted an alternate approach here:

https://www.beatthegmat.com/gmat-prep-ds ... 78673.html

[DanaJ]

The thing is, with angles it matters what's in the middle of the three letters. For instance, RSQ has the point at S, while QRS has the point at R. Those are two completely different angles. It's the same with UST and STU: one has the point at S, while the other has the point of the angle at T.

As to why I'm dividing by 2: the triangle QRS is isosceles, since it has two sides that have the same length. This means that angles RQS and RSQ are are equal. In any given triangle, the sum of angles is always 180 degrees, which means that you have the following relationship:

RQS + RSQ + QRS = 180

But the first two are equal, so:

2RSQ + QRS = 180

so in the end:

2RSQ = 180 - QRS

RSQ = (180 - QRS)/2

Hope this helps!