Thread: Triangle 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?

Originally Posted by sarahh

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?

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.

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

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.

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..

Originally Posted by sarahh

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.

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?

Originally Posted by sarahh

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,





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

is one of multiple ways to combine b and c.

Study Plato's responses as they are beautifully concise.