In the figure shown, what is the value of x?

(1) The length of the line segment QR is equal to the length of line segment RS.

(2) The length of line segment ST is equal to the length of line segment TU.

Correct Answer is C. I thought it was E. I'm not sure how the lengths of line segments indicate the measurement of angles such that I can conclude the measurement for angle x. I understand there are two isosceles triangles as indicated by two statements. Are there three right isosceles triangles including the largest triangle as a whole? Even if so, how can I derive x?

Thanks!

As per the statements 1 and 2, we can form an another triangle by joining Q and C as mentioned in the diagram. so we will have 5 triangles, RPT, RQS, SCT, SQC and PQC. To find x, we need to know the measurements of triangle SQC. Angle x= QSC which is the hyptoneuse for triangle QPC which is right triangle.

Hence C.

Thanks

Thanks for your response Sudhan. I think I'm a little confused as to where C is. How can I join Q & C? Where is C? Is that where S is?

I am sorry. That should be U not C as I mentioned earlier.

Thanks

well, you've got two isosceles triangles if you take the two statements together. so, in other words, angles RQS and RSQ are the same, and angles TUS and TSU are the same.

the big triangle (the one containing everything in the problem) is a right triangle, so you know that angle R and angle T add to 90 degrees. therefore, let angle r be r degrees, and let angle T be (90 - r) degrees.

then
angle RSQ = (180 - r)/2
= 90 - (r/2) degrees
and
angle TSU = (180 - (90 - r))/2
= 45 + (r/2) degrees

so
x = 180 - RSQ - TSU
= 45 degrees.

--

this is also an excellent problem for picking numbers. you can pick any number of degrees you want for angle R (as long as it's acute, of course), and then let angle T be 90 minus that number of degrees. then work your way through the problem, knowing that you can use the isosceles triangles to figure out everything else in the problem.

if you do this with two or three sets of numbers, you'll notice that you get 45 degrees every time. coincidence? not likely.

thank you for that explanation. that was great.

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This problem is insane. Things like this make me say to myself, "a person that would get a 790 would look at these two combined and come up with an answer in 30 seconds. I'm going with C."

Goal is to find x.
If we find ang(RSQ) and ang (UST), we can add them and subtract the result from 180 to get x.

ang (RSQ) = y and ang (STU) = z [say]

ang (RQS) = y and ang (SUT) = y [because RQS and SUT are isoceles]

In RQS, ang (QRS) = 180-2y and in SUT, ang (STU)=180-2z

In RPT, 90 + 180-2y + 180-2z = 180,
=> y+z = 135, => x=45