Lines & Angles Q 1

[aditiniyer]

Please let me know how to solve the attached question.

[GMATGuruNY]

If x - q = s - y, what is the value of z?

1) xq + sy + sx + yq = zr
2) zq - ry = rx - zs

x-q = s-y
x+y = q+s.

Angles inside a triangle must sum to 180.
Since x, y and z must sum to 180, x+y = 180-z.
Since r, q and s must sum to 180, q+s = 180-r.
Substituting x+y = 180-z and q+s = 180-r into x+y = q+s, we get:
180-z = 180-r
r = z.

Statement 1: xq + sy + sx + yq = zr
xq + sy + sx + yq = zr
x(q+s) + y(q+s) = zr
(q+s)(x+y) = zr.

Substituting q+s = x+y on the left side and r=z on the right side, we get:
(x+y)(x+y) = zz
(x+y)² = z²
x+y = z.
Since (x+y) and z must sum to 180 -- and (x+y) and z are EQUAL -- we get:
x+y=90 and z=90.
SUFFICIENT.

Statement 2: zq - ry = rx - zs
zq + zs = rx + ry
z(q+s) = r(x+y)
z/(x+y) = r/(q+s).

It's possible that z=r=10 and x+y = q+s = 170, with the result that the angles inside each triangle sum to 180.
It's possible that z=r=20 and x+y = q+s = 160, with the result that the angles inside each triangle sum to 180.
Since z can be different values, INSUFFICIENT.

The correct answer is A.

[Mo2men]

Statement 2: zq - ry = rx - zs
zq + zs = rx + ry
z(q+s) = r(x+y)
z/(x+y) = r/(q+s).

It's possible that z=r=10 and x+y = q+s = 170, with the result that the angles inside each triangle sum to 180.
It's possible that z=r=20 and x+y = q+s = 160, with the result that the angles inside each triangle sum to 180.
Since z can be different values, INSUFFICIENT.

The correct answer is A.

Dear GAMT Guru

In S2, As (q+s)= (x+y) so we can rule them out and we get z=r which is NOT new info...Hence it is insufficient.

Is my line of reasoning acceptable?

Thanks

[GMATGuruNY]

Statement 2: zq - ry = rx - zs
zq + zs = rx + ry
z(q+s) = r(x+y)
z/(x+y) = r/(q+s).

It's possible that z=r=10 and x+y = q+s = 170, with the result that the angles inside each triangle sum to 180.
It's possible that z=r=20 and x+y = q+s = 160, with the result that the angles inside each triangle sum to 180.
Since z can be different values, INSUFFICIENT.

The correct answer is A.

Dear GAMT Guru

In S2, As (q+s)= (x+y) so we can rule them out and we get z=r which is NOT new info...Hence it is insufficient.

Is my line of reasoning acceptable?

Thanks

Perfect!