Bunuel wrote:
In the diagram above, AC = AB, and angle DAB = angle DBC. What is the measure of angle BCD?
(1) angle BDC = 2*(angle DAB)
(2) AD = BD
Kudos for a correct solution.Attachment:
gdrtq_img4.png
VERITAS PREP OFFICIAL SOLUTION:From the prompt, we know that triangle ABC is
isosceles, with AB = AC and angle ABC = angle DCB. Because angle DAB = angle DBC, and they share the angle at C, we know triangle BCD is similar to triangle ABC; therefore, triangle BCD must also be isosceles, with BC = BD and angle BDC = angle BCD. For simplicity, let’s say that
x = angle DAB = angle DBC
y = angle ABC = angle DCB = angle BDC
We know that (x + 2y) = 180°, and the prompt is asking for the value of y.
Statement #1: angle BDC = 2*(angle DAB)
In other words, y = 2x. Then
x + 2y = x + 4x = 5x = 180°
This means x = 36° and y = 72°. This statement leads directly to the numerical value sought in the prompt. This statement, alone and by itself, is sufficient.
Statement #2: AD = BD
This tell us that triangle ABD is also isosceles. This means that angle DAB = angle ABD. Think about angle ABD. That angle is the “leftover” between two angles we have already discussed:
angle ABD = (angle ABC) – (angle CBD) = y – x
Well, angle DAB = x, so if these two are equal, this means:
y – x = x
y = 2x
This turns out to be the exact same information that was given in statement #1, which we already know is full sufficient.
Answer = (D)
(BTW, more than you need to know for the GMAT, but these are Golden Triangles, because the ratio AB/BC equals the Golden Ratio! ) _________________