In a previous post, where I brainstormed ideas on a sample topic, I mentioned I would write a mock essay using those ideas. Below is that topic. And below that is the essay.

Prompt

In any field of endeavor, it is impossible to make a significant contribution without first being strongly influenced by past achievements within that field.

Write a response in which you discuss the extent to which you agree or disagree with the statement and explain your reasoning for the position you take. In developing and supporting your position, you should consider ways in which the statement might or might not hold true and explain how these considerations shape your position.

Essay

While many accomplishments build on existing knowledge, many significant breakthroughs result from those who have completely broken with tradition. Clearly then, significant breakthroughs do not always build upon existing knowledge.

By the mid-20^th Century, Abstract art was at its acme. At the same time, many critics felt that the artists had all but exhausted themselves in terms of innovation. To create unique one had to be random and chaotic—silly even. It was little surprise that many crowed about the imminent demise of Abstract art. It was into this milieu that Jackson Pollack the artist came into being. Seemingly random and chaotic—though anything but silly—his work had no clear precedent. He would throw paint at a canvas on the ground or propped up against the wall. Though many dismissed him as a madman, a crank, eventually his work was hailed as genius, and that work did much to reinvigorate the movement of Abstract Art. Had Pollack tried to simply build off of the existing art he very well may have fallen victim to the prophecy that the Abstract art movement had become effete and moribund.

Whereas Jackson Pollack created seminal work by totally breaking with tradition, Copernicus questioned tradition, specifically the assumptions underlying the Ptolemaic version of the universe. At the time, the geocentric model of the universe was considered gospel: both the church and the scientific community (at that time the two were heavily interrelated) endorsed this view because it put man at the center of the universe. Even then, it had the stamp of scientific legitimacy in Ptolemy, who had centuries earlier “proven” that the Sun revolved around the Earth. Using advanced math and his observation skills, Copernicus was able to determine that the Sun revolved around the earth. Just as importantly, he was able to challenge the orthodox view, and instead of building off of it, exposed the shoddy foundations upon which this view was built.

Of course both Jackson Pollack and Copernicus are not the typical in the sweep of human endeavors. It is true that many breakthroughs result from somebody building off of or simply improving the way in which something is done. Henry Ford, with his use of the assembly line, made the process of manufacturing goods far more effective. Then there is our modern day visionary, Steve Jobs, who took existing technology and simply made it easier to use (and prettier to look at).

Yet, as Pollack and Copernicus show, there are certain instances in which thinkers are able to completely break with tradition. Therefore, not all significant breakthroughs result from those who build upon previous knowledge. But in those instances in which a thinker has challenged age old wisdom, or come up with a radical form of art, the world surely takes notice.

Analysis

This is by no means a perfect essay. But it effectively develops the point: It is not impossible to come up with a breakthrough that doesn’t build off previous knowledge. This is an example-heavy essay, building its position off of Jackson Pollack and Copernicus. Now I am no Art History major, and I can’t profess that this is example is 100% correct. The thing is the GRE won’t dock your grade if you fudge slight facts. For instance, if I said Copernicus was a 17^th Century astronomer, instead of 15^th Century one, that would not be too catastrophic. If I place him as a contemporary of Jackson Pollack, then I am in trouble. So again, you have a little leeway in facts.

It is by no means a bad idea to make an essay an example-driven one. At the same time, I don’t think there was too much analysis of the prompt. Perhaps had the essay delved more into what it means to make a significant contribution to a field that would have been helpful. For instance, Copernicus didn’t just come up with his heliocentric view out of the blue, uninfluenced by other thinkers. After all he did employ math and observations gleaned from new technology. Had he not applied the math and used a telescope he would have not made his discovery. So was this application, in a sense, building off previous knowledge?

Takeaway

Nonetheless, the essay did a decent job and would probably receive a ‘5.’ This essay is just one possible iteration of a ‘5’. To get a sense of more ‘5’s and ‘4’s, and even ‘6’, you can find sample essays on the gre.org site. My advice is to understand the difference between a ‘4’ and a ‘5’, do your best to identify where you are scoring and, when applicable, try to get to a ‘4’ by identifying your weaknesses, or a ‘5’, of, for some, a ‘6.’

Last month, I shared thoughts from our first customer spotlight. I now wanted to share some of the survey responses we received in May. As a reminder, these are survey responses from Magoosh customers who completed the GRE and were kind enough to provide a picture. One of our customers had over a 300 point improvement based on the old scale! This could be you next month

See the full list of customer reviews here.

GRE words are often frightening because of their length. Who hasn’t buckled at disingenuousness, quaked at pusillanimity, and all but thrown their hands up in the air at obstreperously. Surely, some of the most difficult words on the test must be these multi-syllable monstrosities.

While such words are clearly intimidating, often one has recourse to word roots, or a positive/negative sense of the word. Very short words, on the other hand, can flummox us to an even greater degree. Eke, irk, nub, and din…sound like the names of a band of troglodytes. Yet these syllabic grunts connote a world of meaning – a meaning you would probably never be able to guess based on the way the words look and/or sound.

Below are a few examples of three- and four-letter words that, despite their diminutive stature, can stop you dead in your vocabulary tracks.

Tyro

A tyro is a rookie, a neophyte, a novice. If you have just signed up for the GRE, and have only begun to comb the wealth of material out there, then you are a GRE tyro. No worries—spend some time on the Magoosh blog, and you will in no time become a GRE maven.

Whit

Often confused with ‘wit’, which is the ability to use just a few words to say something funny, ‘whit’ means a tiny amount. Often we encounter this word couched as follows, “not a whit“, which translates to ‘not at all.’ As in:

I care not a whit whether you are a tyro of the comedy stage; you have a natural wit that will win you the affection—and guffaws—of your audience.

Nub

Nub is the central or crucial part to a matter. Nub is related to gist and crux (as well as ‘pith’, which appears below). It is fitting, that in each case, a word meaning the basic overall meaning, is in itself short.

Din

A loud, ongoing noise is a din. Crowds are known for their din. Noisy restaurants, the frenetic kitchens in the back, are all guilty of creating a din. An informal synonym for ‘din’ is ‘racket.’ (And I’m not talking about the thing Federer wields). A GRE synonym for din is the word ‘cacophony.’

Irk

To annoy is to irk. A similar word—and another three-letter word to boot—is vex. Remember to use both correctly

Eke

No, this is not a response to touching some mysterious oozy substance (that would be ick). Eke means to barely manage or subsist on. If you are eking out a living, then you are barely making enough money to get by. If a team ekes out a victory, then they just barely win.

Pith

The core or essence of something is the pith. Matters have piths, as in, “The pith of the matter was the elephant in the room that everyone blithely skirted around.” The pith of this post is the following: be careful with short words; make sure you learn the definitions and can comfortably use the words in a sentence.

For many, coordinate geometry is already a daunting concept. When a question dispenses with the graph all together, students can feel even more at a loss. If you fall into this group, do not despair. Here is a helpful guideline:

Do Not Always Draw the Graph

This advice may seem counterintuitive. After all, the problem didn’t provide a graph. Wouldn’t the first step be to graph the problem out?

Many coordinate geometry concepts sans graph are testing your conceptual thinking. Take the follow problem:

1. Which of the following lines do not contain coordinate points that are both negative?

Solution:

The slope formula is important – if the question is explicitly asking for the slope. What is often more important is knowing that a line with a negative slope – from left to right – slopes downward. A positive slope, unsurprisingly, slopes upward.

Think of it this way – start at a negative x-coordinate (say -2) of a line. If you were to place a ball on the line would roll down the line as it move into positive territory for the x-coordinate? If so the line is negative, if not the line is positive.

For this problem, we are looking for a line that does not pass through the third quadrant – the quadrant in which x and y are both negative. Graph the point (-2, -2). That’s in the third quadrant.

Now here’s the big conceptual part – any line that slopes upwards will always pass through Quadrant III. Graph it if you have to – or simply imagine a line of infinite length sloping upwards. Anyway you try to do so there will always be the Third Quadrant waiting to claim a part of your line.

Now, imagine a downward sloping line. Is it also crossing through the third quadrant? Well, move the entire line to the right. At a certain point, your line will no longer be in the Third Quadrant. As long as that line cross the y-axis at a positive value, it will never cross through the Third Quadrant.

Now you only need to find two things: a line that has a positive y-intercept and a negative slope. And that is much better than having to graph every one of the equations in answer choices A – E!

Only answer (C) x + y = 2, which can be re-written as y = -x + 2, has a negative slope (-1) and positive y-intercept (+2).

Takeaway

If a coordinate geometry question does not provide a graph, it is often testing conceptual thinking. So unless you are really desperate (which can happen on the GRE), and have time to spare, avoid graphing and think conceptually.

If you are unsure what to do, take a step back from the problem and ask yourself: will graphing this problem out take a long time? If the answer is yes, then there is very likely a much faster, no-graph approach.

Some words masquerade—hiding behind a false appearance, ready to deceive and lead us astray. I’m not talking about words that look like other words (e.g., loath and loathe) but words whose definitions themselves speak of duplicity and the counterfeit.

1. Mountebank

150 years ago, about the time that Wyatt Earp was corralling the bad guys, and Tombstone was riddled with fresh tombstones, a certain kind of salesman would make his way across the great expanse of the United States.

In small towns, he would set up shop, mounting a large bench and hawking the most wonderful remedies: mam, this green oil will make your skin glow like a baby’s; and lad, this purple snake’s oil will give you the strength of three men.

By the timeit was clear that the green potion caused nothing more than indigestion, and the purple a nasty case of lockjaw, our huckster would be hundreds miles away, plying his wondrous wares on a fresh crop of dupes.

Remember how I said he would mount a large bench? Well, banco is the Italian for bench, so a mountebank and is one who figuratively mounts a bench and tries to sell us snake oil, or any product with no value effect, especially the one the mountebank touts.

Today, mountebank can refer to anyone who makes false claims in order to extract money from us.

2. Spurious

Okay, spurious doesn’t get a nice fancy back-story like mountebank. Spurious refers to anything that is counterfeit or fake. Claims can be spurious, meaning they are not authentic (see apocryphal, below).

Do not confuse specious with spurious. Though the two words look similar, specious means something that is attractive but ultimately misleading and false. That is not to say that which is spurious is not specious. Sometimes the two overlap. However often the contexts in which they are used are different. For example, an argument can be specious and a claim spurious, but not vice-versa.

3. Nostrum

Oh yes, the years of the mountebank are long behind us. The idea that anyone would try to mulct us of our hard-earned money, hawkingsham remedies right in front of our eyes: pills that imbue us with the sheen of youth; 10-hours of energy in a bottle; Brazilian berries that allow us to leap tall buildings in a single bound.

He may no longer travel in a horse drawn carriage, but the mountebank is very much alive, setting up his shop all over the Internet. And all these unproven remedies have their own word: nostrum.

Properly defined, a nostrum is any medicine or cure that does not live up to its claim. Nostrum can also apply to social remedies—that is plans, philosophies that claim to cure society of its many ills.

4. Veneer

This word describes a surface that is appealing but covers up something negative beneath. Politicians, beneath a veneer of white-toothed smiles, aren’t always trustworthy.

Veneer also describes the small glossy coating on wood that covers the coarse wood below. Makes sense: the polish of the top often belies the coarseness below. Not too different for some politicians.

5. Apocryphal

Quack remedies are often tangible; apocryphal is reserved for the intangible—stories or reports that are of dubious authenticity. And apocryphal stories are not intentionally misleading. Many untrue stories are simply bandied about, with few people every questioning the validity of their respective sources.

In religious circles, this word is a contentious one. The Apocrypha were parts of the Bible that many did not consider authentic. Did Jesus really say and do that, the debate rages.  As a result, the Apocrypha have been let out of the New Testament.

The modern use of the word apocryphal does not typically pertain to a religious context. Today any story that is of dubious validity is apocryphal.

So remember: Always check your sources. Always scratch the veneer to see what lies below. And always watch out for the mountebank and his nostrums. They are masquerading all around us.

By Valeri Boyle

 This is a guest post from GradSchools.com – the Internet’s leading source of information for students seeking a graduate program.  Find your graduate program at GradSchools.com, and visit us on Facebook.

Graduate school. It’s something that almost everyone considers at some point in his or her life. Whether you’re looking to make more money, or want to perform research or teach, there are a variety of reasons to attend grad school.

1.      Necessity.

Some professions, such as Anthropologists, Physician Assistants, Epidemiologists, Psychologists, and Speech-Language Pathologists, require a graduate degree or higher to even begin working the industry. To see the minimum education required in your field, check out the Bureau of Labor Statistics’ webpage.

2.      Stand out from your peers.

The “academic inflation” phenomenon has resulted in an excess of college-educated individuals competing for too few jobs. A graduate degree may help you stand out from your peers in this extremely competitive job market and help ensure that you find a position upon graduation.

3.      Ability to earn a higher salary.

Just because your chosen industry doesn’t require a graduate degree, doesn’t mean they don’t prefer it. Obtaining a master’s degree may allow you to earn a higher salary than if you just had the minimum education needed to enter the profession.

4.      Ability to climb the corporate ladder more quickly.

In many cases, having an advanced degree can allow you to climb the corporate ladder more quickly than those with only a bachelor’s/associates degree. Even if obtaining a graduate degree doesn’t automatically earn you a higher position, it could easily open doors to future promotions and job opportunities.

5.      Service-orientated programs.

Many graduate-level courses are taught as discussion-heavy seminars rather than the lectures you are used to attending as an undergrad. You also have the ability to choose a service-orientated program which requires hands-on experience in the field via an internship or practicum. This can allow you to receive an overall enhanced understanding of the field.

6.      Option of writing a thesis/dissertation, conducting research, or teaching.

Graduate school is much more than just classes; you are able to complete a variety of projects to improve your knowledge of the industry. Many schools require graduate students to write a thesis or dissertation before graduating. This can allow you to study, in detail, a specific aspect of your chosen industry. If your findings get published, you can receive national or even international recognition for your work.

You may also get the option to conduct research while in graduate school. Many schools provide top-of-the-line equipment for students and faculty to perform research. Publishing your research could once again allow you to obtain national or international recognition. Finally, if sharing your knowledge is important to you, many graduate students are given the opportunity to teach a class. Whether it is through a GA or TA position, or just because a professor recognizes your outstanding knowledge of a subject, you may be given the opportunity to teach a class or even an entire course. Who knows, maybe you’ll find that teaching is your passion!

7.      You want to.

While everything listed above are great consequences of attending graduate school, you shouldn’t do it unless you want to. Graduate school is an enormous commitment, and you need to want to put in the time, money and effort it requires.

Seven Reasons to Go to Grad School was originally posted at GradSchools.com.

Proportions are extremely common on the GRE. If you don’t have a strong grasp of them, and you are busy trying to figure out combinations/permutations or probability, stop. Focus your attention on mastering proportions before moving on to more tertiary concepts.

So let’s start basic. Proportions can be broken up into two groups: direct proportions and indirect proportions. In this post I am going to focus on direct proportions. They are more intuitive than indirect proportions and are also the more common on the GRE.

Direct Proportion

Below is an example of a direct proportion:

To solve for x we cross multiply, giving us: ; .

This is a direct proportion because as the ‘5’ becomes larger (namely it quadruples to become 20) the x also gets larger (it quadruples to become 8). That is both sides are getting larger.

I can almost guarantee that you will not see such a straightforward equation on the GRE Quant section. Instead, you will be given either a word problem or a graph and you will have to translate the information into an equation like the one above.

Let’s take a look at two problems:

In 2004, 2,400 condos sold, 15% of the total housing units sold that year. If 25% of the homes sold in 2004 were four-bedroom houses, then how many four-bedroom homes sold in 2004?

(A)  3,600

(B)  4,000

(C)  4,200

(D) 4,800

(E)  6,000

Explanation

Here we want to set up an equation. ; ; . (B).

Some things to note: you can take off the last two zeroes in 2,400 to get 24, a step which will make the math easier. Remember to bring the two zeroes back, which makes sense: x =40, is clearly too low and not amongst the answer choices.

Speaking of answer choices, notice that 25% is less than double of 15%. Therefore, the number of four-bedroom houses sold has to be less than twice the number of condos sold. (D) and (E) cannot be answer. (A) 3600 is only 50% greater than 2,400, so it is probably too low as well. Elimination, esp. if you are short on time, or getting tangled up in the calculation can be very effective.

Now let’s put a spin to this question. Nothing too tricky; indeed you should be able to solve this using the method above.

In 2004, 2,400 condos sold, 15% of the total housing units sold that year. How many units sold in 2004 were not condos?

(A)  16,000

(B)  13,600

(C)  12,400

(D) 11, 200

(E)  8,600

Explanation:

To approach this question, a good idea is to find the number of units in the total market. Then to find the number that are not condos subtract the condos from the total.

My reason for this approach is it is easier to do the math when we are working with 100 vs. 85 (which would be the percent of home that are not condos).

The solution is as follows:

; . Remember to add the two zeroes: 16,000. Now we have to subtract the total condos (2,400) to find the number of units that are not condos: 16,000 – 2,400 = 13,600.

Takeaway:

Setting up a proportion is essential to solving a range of GRE math questions. Make sure you can confidently and quickly solve this question type before more on to more challenging – but less common – concepts.

We have some BIG news. Based on your feedback, we’ve redesigned our video lessons page in the Magoosh GRE product. Let’s jump right into the changes.

New lesson states

Previously, lessons were either unwatched or watched (denoted with blue or grey links). While these states were helpful, they didn’t account for situations in which you watched a lesson but might need additional review later on. We now have three states:

1. Unwatched: These lessons are blue.
2. Watched: These lessons are blue with an eye icon next to the lesson.
3. Finished: These lessons are green with a green check next to the lesson.

With these states, you can better track which lessons you have yet to watch, which ones you may want to watch again, and which ones you have completed or finished. See the screen shot below:

Add Notes

You can now take notes for each lesson. Your notes are private so no other user can see them.

And you’ll see a pencil icon next to any lesson to which you’ve added a note.

Improved Lessons Interface

Last but not least, we’ve revamped the lessons interface:

1. Navigation for additional lessons is now on the right
2. You can see thumbnails and video lengths for each related lesson
3. You can more easily navigate to the next lesson
4. You can mark each lesson as finished

We hope you like the changes! Feel free to leave a comment below to let us know what you think.

Fact: An 8 year old boy who is 4’5″ (53 inches) tall is in the 86th percentile for height for his age.

What on earth does that mean?  Well, the percentile of an individual tells you what percent of the population has a value of a variable is below that individual’s value of the variable.  For example, to say that a 4’5″ 8 year-old boy is in the 86th percentile for height for his age, we are saying: gather together all 8 year-old boys on Earth, and measure their heights; if you sort out all the 8 year-old boys who have a height less than 4’5″, they will comprise approximately 86% of the population.   That boy is taller than 86% of other boys his age – that means he’s in the 86% percentile.

Percentiles is a relatively unlikely topic to see on the GRE, but if it does show up, here are a few handy facts to have up your sleeve.

Biggest and Smallest

A few details to clarify.  The individual with the lowest value of the variable, with the minimum value, is not bigger than anyone, so the lowest percentile, the percentile of the rock-bottom minimum, is the 0th percentile.  If my score is in the 0th percentile, then I am not higher than anyone.

What’s trickier is the maximum score.  If my score is the highest score, I am higher than everybody else, but that’s not the 100th percentile, because in order to be higher than 100% of the population, higher than everyone, I would have to have a score higher than my own score: a paradox!  In fact, for this very reason, there’s no such thing as a 100th percentile.  The person with the highest score is higher than everybody else, but not higher than herself, so she’s in the 99th percentile.  If we are sticking with whole numbers, the 99th percentile is the highest possible percentile.  If we go to decimals, we can get higher with the 99.9th percentile (1 out of a 1000), the 99.99th percentile (1 out of 10000), etc.

Median and Quartiles

The median is the middle of a list: the median divides a list into an “upper half” and a “lower half.”    This means, the median is higher than the lower half of the population, higher than 50%, so the median is the 50th percentile.  Now, we have to be careful here.  On a list with only three members — e.g. {2, 4, 7} — the median is the middle number, here 4, but that number is higher than only one number out of three — so 4 is the 33rd percentile of that list.  In a technical sense, the median is not always the 50th percentile.

In some sense, though, that’s a specious objection.  When there are only 3 members on a list, nobody in their right mind talks about percentiles.  When the total number is less than a few hundred, there’s seldom talk of a percentile.  Percentiles, by their very nature, are a way to make sense of tens of thousands, even millions of individuals.  How many 8 year-old boys are there on Earth?  Who knows, but it’s certainly a very very large number.  That’s where percentiles are used in practice.

When the number of folks in the group is that large, then for all intents and purposes,the median is the 50th percentile.  If you are familiar with the idea of quartiles, then the first quartile is the 25th percentile and the third quartile is the 75th percentile, again, when the group sizes are truly huge.

Practice Questions

1) Sasha took a nationwide standardized test that is graded on a scale from 20 to 60.  Sasha got one of the best scores record on that this test.

Column A              Column B

Sasha’s score            the percentile of Sasha’s score

(A) The quantity in Column A is greater.

(B) The quantity in Column B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.

2) Alice took nationwide standardize test that is graded on a scale from 0 to 100.  Alice the highest score recorded on that this test.

Column A               Column B

Alice’s score             the percentile of Alice’s score

(A) The quantity in Column A is greater.

(B) The quantity in Column B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.

3) A large distribution of score is normally distributed

Column A

score that’s one standard deviation above the mean

Column B

score that has the 80th percentile

(A) The quantity in Column A is greater.

(B) The quantity in Column B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.

Practice Questions Answers and Explanations

(1) B; (2) D; (3) A;

1) We know that Sasha is near the top of the scoring distribution, so that would mean a score with a percentile close to the 99th percentile.  Because of the scoring scale, the score is not going to be above 60, so the percentile is clearly bigger.  Answer = B.

2) Alice got the highest score, so by definition, that’s the 99th percentile.  What we don’t know is: how hard was this test?  What score was the highest score?  If it was a particularly challenging test, it could be that the highest score anyone achieved was only, say, a 73.  In that case, the percentile would be greater.  If, on the other hand, it was possible to get a perfect score, and Alice did in fact do that, then her score of a 100 would be greater than the percentile.  We don’t have enough information to decide.  Answer = D.

3) Here, it might be helpful to brush up on Normal Distribution.  On a normal distribution, it’s always true that 68% of the populations lies within one standard deviation of the mean.  That means, half of that, 34%, lie between the mean and one standard deviation above the mean.  The score that is one standard deviation is higher than the 34% between the mean and one standard deviation above the mean, as well as than the 50% below the mean.  That means, a score that lies one standard deviation above the mean is the 50 + 34 = 84th percentile.  Thus, it’s higher than a score in the 80th percentile.  Answer = A.

1. The quantity in Column A is greater
2. The quantity in Column B is greater
3. The two quantities are equal
4. The relationship cannot be determined from the information given

Many people dread choosing answer choice (D) on Quantitative Comparison (QC) Some feel it may be conceding defeat. Others think that the GRE is trying to trick them by making them pick (D). After all, they think, there must be some pattern that I’m not getting.

The truth is answer (D) comes up often. And to determine whether an answer cannot be determined is actually not too difficult.

#1 Determine a relationship

Say you find an instance, in which the answer is (A) the information in column A is greater.  If that is the case, then the next step is to disprove that.

#2 Disprove that relationship

Meaning, see if you can come up with an instance, either through plugging in different variables, manipulating algebra, or manipulating a geometric figure, in which the answer is not (A). As soon you do that, you can stop. The answer is (D).

If you can’t disprove your answer, then it must be correct: it must be (A), (B) or (C).

1.

Column A	Column B

1. The quantity in Column A is greater
2. The quantity in Column B is greater
3. The two quantities are equal
4. The relationship cannot be determined from the information given

2. Quadrilateral PQRS has three equal sides.

1. The quantity in Column A is greater
2. The quantity in Column B is greater
3. The two quantities are equal
4. The relationship cannot be determined from the information given

Explanations:

Question #1

After choosing a few numbers you should note something: that Column A will always be greater than Column B.

Why? Will, whenever, you have an even exponent, positive or negative, that exponent will always yield a positive number.

Odd exponents, on the other hand, give you a negative output if the base (the number below the exponent) is negative. Remember x has to be negative. So no matter what number you plug in Column A will always be negative, Column B positive. This is a definite not (D). The answer is (A).

Question #2

Three equal sides equal three equal angles. The number we are given in column B is 240, which is pretty large, considering that we only have two angles to work with.

Can we in any way finagle a number so that we can get two angles to equal 239? Well what if we assume the three angles equal 120. Is this valid? No, because in a quadrilateral we have a total of 360 degrees, and if three angles are equal to 120, then we have zero degrees left over.

So it seems that the answer is column (B). This is the step in which we want to do our best to disprove our answer. So is there anyway, using the constraints in the problem, to come up with two angles that sum to 239? If you assume the three angles are 119, then that would give you 238, which is also less.

But don’t give up now. See if you can think outside the box, or the quadrilateral as it were. See, nowhere in the problem does it say that the angle measurements have to be integers. So if the three sides have measures of 119.9, and the smallest angle has a measure of .3, then the sum of two of the large angles would be greater than 238.

Because we disproved our initial answer of (B), the answer to the question is (D).