Question

The angles in a particular triangle are in arithmetic progression, and the side lengths are 4,5,x. What is the sum of all possible values of x?

Answer 1

I don't know what you mean by:
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What is the sum of all possible values of x?
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but all possible values of x must be greater than one and less than 9.

As far as the angles being an arithmetic progression the only triangle that fits is a 30 60 90 triangle but it doesn't have sides in lengths of 4 and 5.