Angles in Data Sufficiency

The GMAT is an exam steeped in logic, deduction and understanding. In order to succeed on the exam, you should be able to look at any given question objectively and determine what it is asking, and where the traps may lie. Now, this is akin to asking you to navigate a labyrinth while avoiding the Minotaur: just because you know the rules, it doesn’t necessarily mean that you will be successful. Taking the labyrinth as a metaphor, how can you rise to the challenge put forth in front of you?

If you could bring only one item with you on this endeavour, what would it be? The first one that comes to mind may be some kind of weapon should you encounter the Minotaur (or even a jetpack), but the tool you may find most helpful is a map. A map of the labyrinth would undoubtedly be helpful in finding the exit, but you’re likely to be so enthralled by it that you run right into the Minotaur! As such, the map can both help you find the exit and lead to your untimely demise, so it is every bit the proverbial double-edge sword you may have been clamoring for at the beginning of the maze.

Maps on the GMAT act in the same way. You can jot down a graphical interpretation of your (self-imposed, 4-hour maximum) prison, but they might lead you willingly into danger as easily as they resolve your dilemma.

Let’s consider the following Data Sufficiency question sans graph:

In an isosceles triangle DEF, what is the measure of the angle EDF?

(1)    Angle DEF is 96 degrees
(2)    Angle DFE is 42 degrees

(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.

Perhaps a quick reminder than angle EDF indicates the angle formed from the vertices starting at E, going through D and ending at F, which means the angle around D. You can use shorthand notation to make things easier for yourself, as I will do henceforth.

Let us evaluate each statement alone, starting with statement (1). Angle D is 96 degrees, and the other two angles are unknown. However, this is an isosceles triangle, which means that the two other angles must be equal to one another. Thus, they must each be (180-96)/2 or 42 degrees. There is no other way this triangle could be isosceles and respect the condition set out in statement 1. It follows that the Statement is sufficient on its own. We are now down to an A-D split (possibly with B.C.)
Should we decide to draw a triangle, it will likely look like this:

Angle E is at the apex, yawning over the other two angles with its slightly larger than 90 degree smugness. Angles D and F are in either corner at a trim 42 degrees each.

Now let’s evaluate statement 2 alone. Angle F is 42 degrees. If you’ve taken the time to sketch out the triangle, you find that this statement works perfectly in conjunction with statement (1), and pushes you (ouch) into thinking that this is sufficient and the correct answer choice is D. However, this feeling is in large part due to the fact that you’ve drawn a map that fits perfectly with both statements. Remember that in data sufficiency, statement (1) no longer exists when evaluating statement (2). This means that you only really know one angle, and the triangle is assuredly isosceles. Everything else is carryover from the first statement evaluated.

Given that the triangle is isosceles, you know that two of the angles must match, so the initial drawing certainly satisfies all the conditions set out in the question. Nonetheless, could another triangle satisfy the same conditions? If one angle is 42, the others can be 42 and 96, or they could be the two angles that match, leaving 42 as odd one out (the third wheel on a triangle?). The two angles D and E could then also be (180-42)/2 = 69 degrees angle each.

Drawing this triangle for visual closure, we get the potential:

This triangle is much closer to an equilateral, with the two side angles matching and the angle at the apex a lot smaller than in the other triangle above. This triangle is as valid as the other one and therefore this statement is insufficient.

Why is one statement sufficient and the other not? It all comes down to the proposed angles. We know the three angles of a triangle must add up to 180 degrees. In an isosceles triangle, if one is known to be over 90 degrees, it necessarily cannot be the one that matches any other angle, short of having negative angles (92 degrees, 92 degrees and -4 degrees!). If the known angle is less than 90 degrees, then there are two distinct possibilities. Let your knowledge of the concepts be your guide, evaluate the statements using logic and not the potential visual trap of a map and you won’t fall for traps on test day (including the clutches of the GMinotaur).

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Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you occasional tips and tricks for success on your exam.  After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.