we can say that
as b=c so we can subtract them from both side
then : A= a+d=90 B= e+b=90 -------> So A=B which is answer C

Good responses. This problem is an example of when plugging in your own number is unfortunately not as fast as doing the algebra

as AC ll FD=> c=b and ∠ACE=d
we can eliminate c and d form both sides, which leaves

A B
a+d-90 90-e-f

now in triangle ACE, a+d+(e+f)=180=> e+f=180-(a+d), now
we can replace the column B

A B
a+d-90 90-(180-(a+b))

A B
a+d-90 a+d-90

column A= column B (C is the answer)

Set up an implied inequality and perform identical operations on each
quantity, grouping variables:

a + d – c – 90 ? 90 – e – b – f
a + d – c ? 180 – e – b – f
a + d – c + e + b + f ? 180
(a + d + e + f) + (b – c) ? 180

In the last step above, only the order of the variables was changed and
parentheses added to group certain terms. Notice that the angle at point C and
point D is the same, as AC and FD are parallel lines intersected by the
transversal CE. So, the first set of parentheses holds the sum of the interior
angles of the biggest triangle ACE, which is 180. Also because AC and FD
are parallel lines intersected by transversal BE, b = c, so b – c = 0 in the
second set of parentheses.

(a + d + e + f) + (b – c) ? 180
(180) + (0) = 180
The two quantities are equal.

A
a + d – c – 90
add f to both sides we get and also 90
a+f-c+d (as a+f=c (exterior angle property))
c-c+d
=>d

B
90 – e – b – f
=> 180+d -180 (as e+b+d=180)
=>d


thus A=B then Answer is (C)

a+f =c (exterior angle property)

now add f both sides

so a+f+d-c-90 ... 90-e-b

d-90 ... 90-e-b

now add 90 both sides

d ... 180-e-b

henc both equal

Since AC and DF are parallel, some facts we know are:

1) the unnamed angle at point C = angle d, which means that in the big triangle, we know that a + d + e + f = 180, since these compose all of the angles of a single triangle
2) angle c = angle b (due to the rule about a line drawn through 2 parallel lines)

So let's look at
Quantity A: a + d - c - 90 vs. Quantity B: 90 - e - b - f

Since b and c are equal, we can just take -b and -c out of each equation and compare without worrying about them:
Quantity A: a + d - 90 vs. Quantity B: 90 - e - f

We know that a + d + e + f = 180
Substract e and f from both sides --> a + d = 180 - e - f
Subtract 90 from both sides -- > a + d - 90 = 90 - e - f
And now we are back to the two quantities we are trying to compare, so we can see they are equal

Answer = C

One doubt: how do we know AC is parallel to FD? The question stem does not provide this info and even the angles do not mention the same degree in the figure given.

Hello from the GRE Prep Club BumpBot!

Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.