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Math Help - Triangle conditions

  1. #1
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    Triangle conditions

    In triangle ABC, if a > b > c which of the following must be true?

    I. 60 < a < 180
    II. 45 < b < 90
    III. 0  < c < 60

    Why is it only I and III? How can I both prove these are correct and disprove II?
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  2. #2
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    Quote Originally Posted by sarahh View Post
    In triangle ABC, if a > b > c which of the following must be true?

    I. 60 < a < 180
    II. 45 < b < 90
    III. 0  < c < 60

    Why is it only I and III? How can I both prove these are correct and disprove II?
    I. If the triangle was equilateral, then all 3 angles would be 60^o

    Hence, if the triangle has a largest angle, it is greater than 60^o

    while all angles must be less than 180^o so that they can sum to 180^o


    III. If one angle is greater than 60^o than another must be less than 60^o.

    This is because the two remaining angles sum to less than 120^o

    and if they were equal, they'd both be less than 60^o.


    II. If a=170^o and c=1^o then b=9^o contradicts the statement.
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    Hmm but how come that would be 2's explanation (the triangle inquality theorem would fail wouldn't it?) Also, I still don't understand why angle a could be 179 (since that value would fit the bounds given--again the triangle inequality theorem would fail for angles b and c.) Maybe if we took some concrete examples? I still don't see it yet
    Last edited by sarahh; December 26th 2010 at 03:29 AM.
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  4. #4
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    The key is in the given, a>b>c.
    If I were false the all three angles are less than 60: impossible.

    If III were false the all three angles are greater than 60: impossible.

    But as Mr. Meade has shown, II can be false without contradiction.
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    But wait, that is just one way to read part I..if a = 179 how could this be? What about an example of: a = 80, b = 70, and c = 30, where all 3 are true above..
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    Quote Originally Posted by sarahh View Post
    But wait, that is just one way to read part I..if a = 179 how could this be? What about an example of: a = 80, b = 70, and c = 30, where all 3 are true above..
    All of that is true. But it has nothing to do with the question.
    The question ask: Which of these conditions must be true?
    You have been shown that I & III must be true but II can be false.
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  7. #7
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    That's kinda my point too, I mean if it has to be true for all possible numbers in the interval then why could a = 179 in the first case?
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  8. #8
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    Quote Originally Posted by sarahh View Post
    That's kinda my point too, I mean if it has to be true for all possible numbers in the interval then why could a = 179 in the first case?
    Hi sarahh,

    a=179^o

    b+c=1^o,\;\;b>c

    I and III are vindicated as normal but II is violated.

    b=0.8^o,\;\;c=0.2^o is one of multiple ways to combine b and c.

    Study Plato's responses as they are beautifully concise.
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