Prove that in a triangle, other then an an eguilateral triangle, angle opposite the longest side is greater than `2/3` of a right angle

Explanation

Consider `Delta ABC ` in the longest side.
`angle A =2/3 `right angle
Proof in `Delta ABC " " BC gt AB.`
[Consider BC is the largest sid ]
`rArr angel A gt angle C`
[angle opposite the longest side is greatest ]
and `BC gt AC `
`rArr angleA gt angle B`
[angle opposite the longest side is greatest]
On addings Eqs ,(i) and (ii) we get
`2angleA gt angle B + angle C `
` rArr 2 angle A + angle A gt angle A +angle B +angle C` [adding `angle A` both side ]
` rArr 3 angle A gt angle A +angle B +angle C`
` rArr 3 angle A gt 180^(@)` [Sum of all the angles of a triangle is `180^(@)`]
`rArr angle A gt (2)/(3)xx 90^(@)`
` angle A gt (2)/(3)` of a right angle