Question 20 Prove that in a triangle, other than an equilateral triangle, angle opposite the longest side is greater than 23 of a right angle.
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ΔABC in which BC is the longest side. In ΔABC,BC>AB, [consider BC is the longest 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 Eqs. (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 angles of a triangle is 180∘] ⇒∠A>23×90∘ i.e.,∠A>23ofarightangle.