Question 495264: prove that in a triangle, other than an equilateral triangle, angle opposite to the longest side is greater than 2/3 of a right angle.
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! "2/3 of a right angle" is just some tricky way of saying 60 degrees (ironically).
It follows from the law of sines that the angle opposite to the longest side will be the largest angle. By the Pigeonhole principle, if we were to distribute 180 "degrees" randomly into three "angles," at least one of the angles would have to be at least 60 degrees. However, the largest angle cannot be exactly 60 degrees, so it must be more than 60 degrees.
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