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Home / Geometry / Prove that the angle opposite the greatest side of a triangle is also the greatest angle of that triangle.

Prove that the angle opposite the greatest side of a triangle is also the greatest angle of that triangle.

Problem: Prove that the hypotenuse of a right angled triangle is the greatest side.
Solution: General enunciation: We have to prove that the hypotenuse of a right angled triangle is the greatest side.

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Particular enunciation: Let ∆ABC be right angled triangle in which ∠ABC = right angle or 900 and AC be the hypotenuse. It is required to prove that AC be the greatest side.

Proof: We know, in a right angled triangle one angle is right angle and the other two angles are acute angles.

∴ ∠ABC> ∠ACB

We know, the side opposite to the greater angle is greater.

∴ AC>AB

Again, ∠ABC> ∠BAC

       ⟹ AC>BC

∴ Side AC be greater than the sides AB and BC.

Therefore, AC be the greatest side. (Proved)

Problem: Prove that the angle opposite the greatest side of a triangle is also the greatest angle of that triangle.

Solution: General enunciation: We have to prove that the angle opposite the greatest side of a triangle is also the greatest angle of that triangle.

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Particular enunciation: Let the greatest side of ∆ABC is AC. It is required to prove that ∠ABC is the greatest angle of the triangle.

Proof: In triangle ABC, AC>AB

We know, the angle opposite to the greater side is greater.

∴ ∠ABC> ∠ACB

Again, AC>BC

       ⟹∠ABC> ∠BAC

Hence ∠ABC be greater than ∠ACB and ∠BAC.

Therefore, ∠ABC be the greatest angle of the triangle.

Problem: Prove that difference of any two sides of a triangle is less than the third side.

Solution: General enunciation: We have to prove that difference of any two sides of a triangle is less than the third side.

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Particular enunciation: Let, in ∆ABC, AB is greatest side. Prove that difference of any two sides of a triangle is less than the third side.

Construction: By joining C , D.
Proof: We know, sum of two side of any triangle is greater than third side.
AC+BC>AB
⟹ AC+BC-AC>AB-AC
⟹BC>AB-AD
⟹BC>BD
⟹BC>BC
So, Difference of any two sides of a triangle is less than the third side.

(Proved)

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