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GMAT Data Sufficiency: Congruence Rules

Learn how to use these formal Geometry rules to your advantage on the GMAT Data Sufficiency section!

 

A reminder about Data Sufficiency

If this question format is new to you, read this post.  If you have done DS questions for a while, it’s still worthwhile to remind yourself of the basics.  You don’t actually need to find a numerical answer on DS.  All you need to determine is: could I find the answer?  Is there enough information to determine an answer?  That’s always the real question on DS.  It is always helpful to remind yourself of that curious shift in perspective each time you start working DS again.

 

Congruence rules

Now, a trip down memory lane.  In Geometry, which you probably took early in your high school career, you may remember the four congruence rules.  These are rules that absolutely guarantee that two triangles are congruent: if two triangles share this particular combination of traits, there is not any possibility that they could differ in the least.  These four congruence rules are known by convenient three letter abbreviations.  (Does this ring a bell now?)

I.  SSS = Side-Side-Side = if you know the lengths of all three sides of a triangle, that uniquely determines all three angle as well as the shape & size of the triangle.  Only one triangle can have those three sides.

II. SAS = Side-Angle-Side (i.e. two sides and the angle between them).  Here, you know two of the three sides, and the angle you know is the angle between the two sides.  This combination of information is enough to determine unambiguously everything else about the triangle: the other side, the other two angles, the size and the shape.  Only one triangle can have this particular combination.

III. ASA = Angle-Side-Angle (i.e. two angles and the side between them).  Here, you know two of the three angles, and the side length you know is the one connecting those two vertices.  This combination of information is enough to determine unambiguously everything else about the triangle: the other angle, the other two sides, the size and the shape.  Only one triangle can have this particular combination.

IV. AAS (a.k.a. SAA) = Angle-Angle-Side (i.e. two angles and the side not between them). Here, you know two of the three angles, and the side length you know is one of the two sides not connecting those two vertices.  This combination of information is enough to determine unambiguously everything else about the triangle: the other angle, the other two sides, the size and the shape.  Only one triangle can have this particular combination.

AAS Triangle

If you have a combination of information that fits into any of those four categories, then that combination uniquely determines the shape and size of the triangle.  Every facet of a triangle is fully determined by any of these four combinations.

 

The ones that don’t work

It’s worth taking a moment just to be clear about the similar sounding combinations of letters, the ones that could be confused with congruence rules, but which actually don’t work.

One is AAA (angle-angle-angle).  As it turns out, this is enough to guarantee the shape of the triangle, but because no side is specified, we could scale that shape to any size.

The other involves two sides, and the angle not between those two sides.  (I have confidence in your ability to create a three-letter mnemonic to remind yourself that this doesn’t work!)  Why this doesn’t work is a tricky issue.  The diagram below may illuminate why angle-side-side doesn’t work as a congruence rule, because two different shape triangles can share this particular combination of information.

Even if you don’t understand that diagram, it’s enough to know that this combination doesn’t work — it doesn’t guarantee congruence.

 

The connection to Data Sufficiency

OK, great, those are the four congruence rules, but what would they have to do with DS questions on the GMAT?  Well, for a start, here’s a PS problem that would never appear on the GMAT.

1. In the triangle above, EG = 5, EF = 3, and FG = 7.  What is ∠G?

I won’t even bother listing answer choices.  The point is: with GMAT math, you simply cannot do this problem.  It requires trigonometry to solve, and that’s well beyond what you need to know for the GMAT.  That question is 100% out of bounds for the GMAT.

Now, consider this unlikely but hypothetical question, suspiciously similar to #1.

2. In the triangle above, EG = 5 and EF = 3.  What is ∠G?

Statement #1: FG = 7

Statement #2: ∠E = 120°

Curiously, even though we can’t solve anything, we can still answer the question.  The prompt gives us two sides of the triangle.

Statement #1 gives us the third side: that would result in SSS, which uniquely determines a triangle.  Angle G is now fully determined.  We don’t have the means at our disposal to find that number, but we can say with confidence: whatever G is, is now fully and unambiguously determined by this information.  Statement #1 is sufficient to determine a unique value for ∠G.

Now, forget statement #1.  Statement #2 gives us the angle between the two sides given in the prompt, so we now have SAS, which also uniquely determines a triangle.  Angle G is now fully determined.  We don’t have the means at our disposal to find that number, but we can say with confidence: whatever G is, is now fully and unambiguously determined by this information.  Statement #2 is sufficient to determine a unique value for ∠G.

Both statements are sufficient, so Answer = D.  Even though every single relevant calculation was absolutely beyond what we could do with GMAT math, we could still answer the DS question.  That’s very important!

Even though a DS question just like this would be “fair game” for GMAC to use, it’s unlikely they would actually give a question like this.  Even though you don’t need to solve for numerical answers to figure out a DS question, they almost invariably give questions which could be solved numerically using just GMAT math techniques, so this question is out.  Nevertheless, this demonstrates a hugely important strategy point: the congruence rules could provide a ridiculously efficient geometry shortcut on the DS.

Typically, triangle questions of this genre would involve triangles you solve, for example, the special right triangles.  Suppose, say, the DS question give you enough information to determine that it’s a 30-60-90 triangle, but either you forget the details of that pattern, or you simply don’t want to go through those calculations if you don’t need to do them.  The congruence rules can be an express lane to answering the DS question, a lane that avoids doing any unnecessary calculations — which, of course, should always be your strategy on GMAT Data Sufficiency.  If, at any step, the information given matches on of the four congruence rules, then everything about the triangle is thereby determined —- in other words, you automatically have sufficiency.

By the way, for the geometrically inquisitive who do want to know the value of ∠G from those two problems:

Of course, I give that purely for the entertainment of folks who remember their trigonometry.  Understanding any of that is absolutely irrelevant to the GMAT.

 

Practice problem

Try to apply the congruence rules in reasoning through this problem.

3) http://gmat.magoosh.com/questions/1017

 

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8 Responses to GMAT Data Sufficiency: Congruence Rules

  1. Mohua January 20, 2015 at 3:12 am #

    This post was very helpful.

    • Mike MᶜGarry
      Mike January 20, 2015 at 10:27 am #

      Dear Mohua,
      I’m very glad you found this helpful! 🙂 Best of luck to you in the future!
      Mike 🙂

  2. Menno October 27, 2014 at 12:17 pm #

    I believe image #3 (for the Angle-Angle-Side triangle) incorrectly displays the AAS triangle as it shows the known side as indeed being between the two angle vertices.

    The incongruence between the description in the text and the image can only be a source of confusion and I recommend it be rectified.

    • Mike MᶜGarry
      Mike October 27, 2014 at 1:06 pm #

      Dear Menno,
      I’m happy to help. 🙂 My friend, I believe you were matching that diagram with the wrong description. The illustrative triangle follows the the verbal description, so the third triangle, with ASA, there was demonstrating point III, the ASA case. For some reason, there was no diagram for the AAS case, so I added one here.
      Does all this make sense?
      Mike 🙂

      • Menno October 27, 2014 at 1:35 pm #

        Ah yes, thanks, well done. Mighty helpful article all in all.

        • Mike MᶜGarry
          Mike October 27, 2014 at 1:57 pm #

          Dear Menno,
          You are quite welcome! 🙂 Best of luck to you!
          Mike 🙂

  3. Faruk September 12, 2012 at 8:01 am #

    Hi Mike,
    nice article .:).Usually how many questions from Geometry do appear on the GMAT?

    • Mike MᶜGarry
      Mike September 12, 2012 at 11:27 am #

      Faruk,
      On the 37 questions of a GMAT Quant section, probably about 3-5 would be about Geometry. It’s not a huge topic, but a few Geometry questions will appear on almost every GMAT.
      Mike 🙂


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