In triangle ABC, if which of the following must be true?

I.

II.

III.

Why is it only I and III? How can I both prove these are correct and disprove II?

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- December 26th 2010, 02:47 AMsarahhTriangle conditions
In triangle ABC, if which of the following must be true?

I.

II.

III.

Why is it only I and III? How can I both prove these are correct and disprove II? - December 26th 2010, 03:11 AMArchie Meade
I. If the triangle was equilateral, then all 3 angles would be

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

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

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

This is because the two remaining angles sum to less than

and if they were equal, they'd both be less than

II. If and then contradicts the statement. - December 26th 2010, 03:16 AMsarahh
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 :(

- December 26th 2010, 03:30 AMPlato
The key is in the given, .

If**I**were false the all three angles are less than :**impossible**.

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

But as Mr. Meade has shown,**II**can be false without contradiction. - December 26th 2010, 03:45 AMsarahh
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..

- December 26th 2010, 04:08 AMPlato
- December 26th 2010, 04:44 AMsarahh
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?

- December 26th 2010, 05:12 AMArchie Meade