Question

What is the value of $x+y$ in the figure above?
(1) $w=95$
(2) $z=125$

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

Answer: (C)

Both statements together are sufficient, but neither statement alone is sufficient.

• In the figure above, $a,b,c,$ and $d$ are the degree measures of the interior angles of the quadrilateral formed by the four lines and $a+b+c+d=360$. Then, $w+x+y+z=(180-a)+(180-d)+(180-a)+(180-b)$, which is equal to $720-(a+b+c+d)=720-360=360$. Determine the value of $(x+y)$.
  

1. Given that $w=95$, then $95+x+y+z=360$ and $x+y+z=265.$ If $z=65$, for example, then $x+y=200.$ On the other hand, if $z=100,$ then $x+y=165$; NOT sufficient.
2. Given that $z=125$, then $w+x+y+125=360$ and $w+x+y=235.$ If $w=35,$ for example, then $x + y = 200$. On the other hand, if $w=100$, then $x+y=135;$ NOT sufficient.

• Taking (1) and (2) together, $95+x+y+125=360,$ and so $x+y=140.$ Therefore, (1) and (2) together are sufficient. The correct answer is C; both statements together are sufficient