First, notice that each statement tells you essentially the same thing. If you know that QPR = 30, you know the sum of the other two angles must be 150 (they have to add to 180), and vice versa. If the two statements give you identical information, the only possible answers are D and E, since there could never be any additional value that comes from combining the two statements.

Now, you can see the answer almost immediately if you look at the problem in the right way. Look just at triangle PRS. If you make a new triangle PQS by widening the angle at P by 30 degrees and leaving the angle at S alone, the other angle (at Q) must get 30 degrees narrower, because the angles in the triangle must add to 180. In other words, if one angle stays the same and one angle increases by 30, the remaining angle has to go down by 30.

Or you can do this algebraically:

Call angle RPS 'x'.
Then PRS is 90-x.
Then QRP = 180 - PRS = 180 - (90 - x) = 90 + x
If QPR = 30, then QPR + QRP + PQR = 180 --> 30 + (90 + x) + PQR = 180
--> PQR = 60 - x = PRS - 30.

So PRS is 30 degrees larger than PQR.

D.

Let's try that explanation again!

Look at the smaller right angled triangle, PRS. Call the angle RPS x, and the angle PRS y. Then the angles are:

x, y, 90

Of course, x+y must equal 90 (sum of angles in a triangle is 180). Now look at the larger right angled triangle PQS. We know, from either statement, that QPR is 30. So angle SPQ is x+30. The angles in PQS are:

x+30, ?, 90

Again, these must add to 180, so ? must be y-30. Hope that's more clear.

jimmytwoshoes wrote:Hey, Sorry -

When you say "angle QPS", you are talking about the degree measurement of angle P, correct?

Sorry Jimmy- didn't see this question until now. Angle QPS is the angle you get at P when you connect the points Q, P and S. There are, of course, three different angles at P in the diagram- QPR, QPS and RPS, which is why we need to use the three letters to refer to the angle.