Question

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

Answer 1

Consider: ΔABC in which BC is the longest side.

To prove: ∠A = `2/3` right angle

Proof: In ΔABC, BC > AB   ...[Consider BC is the largest side]

⇒ ∠A > ∠C  ...(i)  [Angle opposite the longest side is greatest] 

And BC > AC

⇒ ∠A > ∠B  ...(ii) [Angle opposite the longest side is greatest]  

On adding equation (i) and (ii), we get

2∠A > ∠B + ∠C

⇒ 2∠A + ∠A > ∠A + ∠B + ∠C   ...[Adding ∠A both sides] 

⇒ 3∠A > ∠A + ∠B + ∠C

⇒ 3∠A > 180°  ...[Sum of all the angles of a triangle is 180°]

⇒ ∠A > `2/3 xx 90^circ`

i.e., ∠A > `2/3` of a right angle

Hence proved.