I know this is about similar triangles but not exactly sure how.Is angle QPS=RPS?
I don't think it's about similar triangles. It is about relationships of angles within triangles.
The stem first tells you that you are comparing angle PQR with PRS. It is essentially asking you, what information is required to be able to compute the difference in measurement of the angles.
First, you are told that QPR is 30 and you know that QSP & RSP are right angles.
The degree of PQS = 180 - 30 [given in 1)] - RPS - 90[RSP]. I assigned variables to the angles like the following:
a = PQS (unknown)
b = PSR (90 degrees)
x = RPS (unknown)
d = PRS (unknown)
Don't ask my why I chose those letters. It's kind of random, but all you need is some way to make it easier to do the math.
So we are trying to figure out if we can compare "d" with "a" when we know that QPS = 30 degrees.
Statement 1 is sufficient and here is why.
Using the variables assigned above you get the following equation:
a = 180 - 30 - x - 90 [180 for all interior angles of a triangle. 30 is already told to us, 90 for the right angle, and x for the unknown portion].
d = 180 - 90 - x. We can go right to answering "how many degrees greater is PRS [d] than PQR[a]".
d - a answers this question, but substituting in for the equation above we get:
(180 - 90 - x) - (180 - 30 - x -90) = V [V for value. I find it easier than just leaving = ____]
90 - x - 60 - x = V
30 - 2x = V
30 = 2x
15 = x
Now plug 15 for x back into the picture and we only have a & d left to solve.
QPS = 45 (30 given in statement 1 and 15 from our solution above)
Angle d = 180 - 90 - 15 = 75
Angle a = 180 - 90 - 30 - 15 = 45
So d - a = 30 degrees. We've done more than is truly required. You coudl stop once you realize statement 1 is sufficient.
Statement 2 is essentially telling you the same thing as Statement one, in a different way.
If you know PQR + PRQ = 150, that is 2 of 3 interior angles. so the 3rd angle has to be 180 - the sum of those 2 (180 - 150 = 30). So you just got through solving when QPR = 30 so you know statement 2 itself is sufficient.
Hope this helps
J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.