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I am a high schooler studying for the SAT, and I came across this question.

In isosceles triangle $ABC$, side $\overline{BC}$ is longer than the other two sides. If the degree measure of $\angle A$ is a multiple of 22, calculate the greatest possible measure of $\angle C$.

(A) 78
(B) 88
(C) 75
(D) 86
(E) 79

The correct answer according to the study guide is (E) 79.

But in the very first sentence of the question, it is stated that side $BC$ is longer than the other two sides, and we know that sides $BA$ and $AC$ are equal in length because this is an isosceles triangle.

That means that $\angle A > 60°$.

Were they just throwing in the part about $BC$ being longer to confuse me?

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    $\begingroup$ point of order: this is a question from a study guide. This is not an SAT question. $\endgroup$
    – Sparr
    Sep 10, 2014 at 17:55
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    $\begingroup$ To me it looks like the mistake they made was mixing up A and C in the question. If you're given that ∠C is a multiple of 22 then (B) is a correct answer. It also generally makes it a more interesting question, with no unnecessary information. $\endgroup$
    – Niall
    Sep 10, 2014 at 22:02

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Your claim that $\triangle ABC$ is obtuse is mistaken. Suppose $\angle A$ were a right angle, for example. Then you would have an isosceles right triangle, and $BC$, the hypotenuse, would certainly be the longest side.

Observe that $BC$ will be equal to the other two sides if the triangle is equilateral—that is when $\angle A=60^\circ$. So $BC$ will be the longest side whenever $\angle A$ is larger than $60^\circ$.

But it seems to me that the problem is still insoluble: the correct answer has $\angle A = 66^\circ$ and so $\angle C = 57^\circ$, which is not a choice.

The proposed solution of $\angle C = 79^\circ$ is clearly wrong. This makes $\angle A = 22^\circ$, and then $BC$ is not the longest side as stated.

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    $\begingroup$ Yes. I believe the were trying to give extra data to confuse the problem not realizing that they were fundamentally changing the problem. $\endgroup$
    – David Greydanus
    Sep 10, 2014 at 15:10
  • $\begingroup$ That is possible, but I think it's more likely that the book answers were misprinted. $\endgroup$
    – MJD
    Sep 10, 2014 at 15:10
  • $\begingroup$ If this was an isolated incident, I would agree but the truth is that this is not an uncommon occurrence. $\endgroup$
    – David Greydanus
    Sep 10, 2014 at 15:12
  • $\begingroup$ Is it possible that this is a "False answer"? I've read that many tests are created with false answers like this, where the test marks the wrong answer as correct (with or without an actual correct answer available). The goal being to watch for consistently wrong answers. If everyone - or a high percentage - answers this question with 79, then the proctor can deduce that the answer key is out in the wild. ("False answers" would be ignored in the actual grading part. not sure what else to call it, if this actually the case or how common this would be). $\endgroup$
    – WernerCD
    Sep 10, 2014 at 17:57
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    $\begingroup$ I think it's more likely that they mixed up A and C in the question. This allows for a correct answer and makes all given information relevant. $\endgroup$
    – Niall
    Sep 10, 2014 at 22:08
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It seems to me that you are right! I think the book got things confused.

Indeed If $BC$ has the longest length, angle A should be the largest, with the other $2$ being smaller (and equal to each other as the triangle is isosceles).

However, the argument you wrote down has a significant error. Just because $BC$ is the longest side, it is not implied that the triangle is obtuse. Consider a triangle with a $70^\circ$ and two $55^\circ$ angles. This triangle is clearly isosceles with two equal angles and yet, not obtuse as all angles are less than $90^\circ$.

I think what they meant is that $BC$ is the smallest side. This will allow for $A$ to be $22^\circ$ and the other two angles to be $79^\circ$.

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    $\begingroup$ I think this is the best answer here, because it's the only one that identifies a simple and plausible error that would result in the book solution being correct. Thanks for posting it. $\endgroup$
    – MJD
    Sep 10, 2014 at 15:38
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If BC is longer than the other 2 sides, then the other 2 sides must be equal because the triangle is isosceles. If indeed they claim that the answer is E, that implies that angle C (79°) is bigger than angle A and thus side AB must be bigger than side BC. So this is clearly incorrect. This is not the first time I have come across such a question from ACT, SAT

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I actually ran into one these type of questions on the real SAT years ago when I took it. The way it was explained to me and my fellow students taking the test was that these questions are generated for the test. They have a set sentence structure and words are pulled from a list to fill in the blanks. In this question we can easily identity the problem. "Side $BC$ is longer than the other two sides" The word longer is the problem here. If you were to change the word "longer" to "shorter" the question would be correct and the answer $E$ $79^\circ$ would also be the correct answer. You get this by knowing that angles $b$ and $c$ are equal and $a$ is a multiple of $22$. All three angles must add up to $180^\circ$ in all.
That means $c+b+(22a)=180^\circ$ but $c$ and $b$ are the same value so $2b+22a = 180^\circ$ which can be rearranged to $b=\frac{180^\circ-22a}{2}$.
Now some may say this equation cannot be solved easily but that is not true. The question is asking for the greatest measurement that angle '$c$' can be. That means that the angle $a$ is as small as it possibly can be. Thus we plug in $1$ for '$a$' giving us $22\times 1$ which is $22$. We use $1$ because $1 \times 22^\circ$ is the smallest possible angle that is a multiple of $22$. We can then solve a rather simple equation of $a b = \frac{180^\circ-22^\circ}{2} = 79^\circ$ the answer.
So in conclusion this question was auto generated and simply pulled one wrong word from a list. This is a kind of a large problem that hasn't really been addressed for $10+$ years.

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    $\begingroup$ I don't think it's true that the questions on the SAT are auto-generated and are not vetted by examiners. Consider the verbal section: auto-generating the reading assessment questions is currently infeasible. So they must be manually generated. It is reasonable to assume that they are double-checked by reviewers at Collegeboard. The same should go for the mathematics section. In fact, the SAT is quite carefully composed in terms of questions and their difficulties. I find it difficult to believe the auto-generation story, esp. considering how controversial it would be if it were true. $\endgroup$
    – Newb
    Sep 11, 2014 at 4:18
  • $\begingroup$ I believe they are suppose to be vetted by examiners. However like all humans they can miss things. I'm sure after going through thousands of booklets their eyes are pretty tired. I am not sure about the reading section but I know for sure the a majority of the math questions are generated in this fashion (they were 7 years ago when I took the SAT). In addition as a programmer, I work for a company that administers tests in various professional fields. We handle a fraction of the number of test that collegeboard does and ALL of our test booklets are generated $\endgroup$
    – keneda
    Sep 11, 2014 at 18:58

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