Choice highlighted with box is OA.
Please help on these DS questions!
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Firstly, please post a single question per thread to avoid confusion.
Now, coming back to the original question:
St1:
QR = RS
This is not at all helpful and in no way helps to find x (cant use similar triangles property, etc)
INSUFFICIENT
St2:
ST = TU
This is not at all helpful and in no way helps to find x (cant use similar triangles property, etc)
INSUFFICIENT
St1+St2:
In the bigger right angled âˆ†RPT the sum of the angles R + T = 180  90 = 90
But In the isosceles âˆ†RQS, R + a + a = 180 ; R + 2a = 180 ; R = 180  2a
Similarly, In the isosceles âˆ†STU, T + b+ b = 180 ; T + 2b = 180 ; T = 180  2b
Thus, 180  2a + 180  2b = 90
We can find a + b.
Now a + x + b = 180 and thus we can find x
SUFFICIENT
Answer C
Now, coming back to the original question:
Q: x = ?In the figure shown, what is the value of x ?
(1) The length of 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.
St1:
QR = RS
This is not at all helpful and in no way helps to find x (cant use similar triangles property, etc)
INSUFFICIENT
St2:
ST = TU
This is not at all helpful and in no way helps to find x (cant use similar triangles property, etc)
INSUFFICIENT
St1+St2:
In the bigger right angled âˆ†RPT the sum of the angles R + T = 180  90 = 90
But In the isosceles âˆ†RQS, R + a + a = 180 ; R + 2a = 180 ; R = 180  2a
Similarly, In the isosceles âˆ†STU, T + b+ b = 180 ; T + 2b = 180 ; T = 180  2b
Thus, 180  2a + 180  2b = 90
We can find a + b.
Now a + x + b = 180 and thus we can find x
SUFFICIENT
Answer C

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Second question is already discussed here: www.beatthegmat.com/distancebetweenx ... 71155.html
Q: We need to find out whether z lies between x and y or not
St1:
xyz < 0
product is negative so either all could be ive or two could be ive and one could be +ive
There could be many possibilities with these combinations and z MAY or MAY not lie inside:
INSUFFICIENT
(All the possibilities need not be visualized here only a few should suffice)
St2:
xy< 0
Either x = +ive & y = ive
or x = ive & y = +ive
We know nothing about Z and z could lie inside or outside.
INSUFFICIENT
St1+St2:
Combining the two inequalities we know that Z is +ive
Case1 (x = +ive & y = ive & Z = +ive) :
Case2 (x = ive & y = +ive & Z = +ive):
More than one possibility, thus INSUFFICIENT
[spoiler]Answer : E[/spoiler]
Given: Distance between x&y is GREATER than the distance between x&z, ie. on a number line xy is a bigger line segment and xz is a smaller one.On the number line, the distance between x and y is greater than the distance between x and z. Does z lie between x and y on the number line?
(1) xyz<0
(2) xy< 0
Q: We need to find out whether z lies between x and y or not
St1:
xyz < 0
product is negative so either all could be ive or two could be ive and one could be +ive
There could be many possibilities with these combinations and z MAY or MAY not lie inside:
INSUFFICIENT
(All the possibilities need not be visualized here only a few should suffice)
St2:
xy< 0
Either x = +ive & y = ive
or x = ive & y = +ive
We know nothing about Z and z could lie inside or outside.
INSUFFICIENT
St1+St2:
Combining the two inequalities we know that Z is +ive
Case1 (x = +ive & y = ive & Z = +ive) :
Case2 (x = ive & y = +ive & Z = +ive):
More than one possibility, thus INSUFFICIENT
[spoiler]Answer : E[/spoiler]