Geometry Question

[RBF3]

I took an educated guess on this question but I was wondering why C was the answer?




Thanks for any help

[RBF3]

Meant to post this in data sufficiency, sorry.

[knight247]

Does look a little complicated but a fairly simple one. Explanation as follows.

(1)in triangle RSQ
qr=rs
So it follows that
angle rqs= angle rsq = y degrees (in a triangle, sides opposite congruent angles are equal)

Now, measure of angles qrs+rqs+rsq=180 (property of triangles)
qrs+y+y=180
qrs=180-2y.......(a)

Insufficient.

(2)in triangle STU
ST=TU
angle sut = angle tsu = z degrees (in a triangle, sides opposite congruent angles are equal)

similarly
measure of angles tsu+sut+stu=180 (property of triangles)
z+z+stu=180
stu=180-2z......(b)

Insufficient

Combining (1) and (2)

In triangle RPT

angle prt + angle rtp + angle rpt =180

180-2y+180-2z+90=180 (since angle qrs=angle prt, angle stu=angle rtp and from 1 and 2)
solving we get
y+z=135.....(c)

now,
angle rsq+angle qsu+angle tsu=180
y+x+z=180

x=180-(y+z)
x=180-135=45

Hence C. Hope this explanation helps.

Have added an image to clarify

[Anurag@Gurome]

I took an educated guess on this question but I was wondering why C was the answer?




Thanks for any help

Note that the measure of angle x depends upon the position of points Q, S and U only. Unless we don't know the fixed positions of these three points, we cannot uniquely determine the measure of angle x.

Statement 1: QR = RS
Thus position of Q and S is fixed. But U can be any point on PT and accordingly value of x will be different.

Not sufficient

Statement 2: ST = TU
Thus position of S and U is fixed. But Q can be any point on PR and accordingly value of x will be different.

Not sufficient

1 & 2 Together: Now the three points are fixed. Let's see whether we can find x. Refer to the image below.


On point S, the sum of the three angles must be equal to 180°.
Thus, (x + y + z) = 180° ..................................... (i)

angle PQS = (180° - angle RQS) = (180° - z)
angle PUS = (180° - angle TUS) = (180° - y)

Now in quadrilateral PQSU,

• Sum of all the internal angles = 360°
  => [x + 90° + (180° - y) + (180° - z)] = 360°
  => (x - y - z + 90°) = 0 .................................. (ii)

Now add (i) and (ii) => (2x + 90°) = 180° => x = 45°

Sufficient

The correct answer is C.