Question 475414: prove that a triangle must have two acute angles
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! The simplest way is to prove it via a contradiction. Suppose that these two angles were not acute, and were both right or obtuse angles. Then the sum of these two angles is at least 180 degrees, contradiction.
Or we can do a Pigeonhole principle proof. Suppose we randomly distribute 180 candies into three boxes. By Pigeonhole, one of the boxes must contain >= 60 candies. If the number of candies in this box is between 60 and 89(technically 89.999...) we can remove this and distribute k candies (k is at most 120) into two boxes and argue again via Pigeonhole that one of these boxes contains less than 90 candies. Or, if the first box contains more than 90 candies, we can distribute the remaining k candies (k is at most 90 here) into the remaining two boxes, both must contain less than 90, so we're done.
I used candies as an example to show how we can simply choose how many degrees are in each angle of a triangle because for any three angles adding up to 180, there is always a unique triangle.
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