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