Question

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

Answer 1

Consider △ PQR where PR is the longest side

So we get PR > PQ

i.e. ∠ Q > ∠ R ….. (1)

We also know that PR > QR

i.e. ∠ Q > ∠ P ….. (2)

By adding both the equations

∠ Q + ∠ Q > ∠ R + ∠ P

So we get

2 ∠ Q > ∠ R + ∠ P

By adding ∠ Q on both LHS and RHS

2 ∠ Q + ∠ Q > ∠ R + ∠ P + ∠ Q

We know that ∠ R + ∠ P + ∠ Q = 180^o

So we get

3 ∠ Q > 180^o

By division

∠ Q > 60^o

So we get

∠ Q > 2/3 (90^o)

i.e. ∠ Q > 2/3 of a right angle

Therefore, it is proved that in a triangle, other than an equilateral triangle, angle opposite the longest side is greater 2/3 of a right angle.