Is the sum of any two sides of a triangle greater than the third side?
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TheoremGiven a triangle $ABC$, the sum of the lengths of any two sides of the triangle is greater than the length of the third side.
(The Elements: Book $\text{I}$: Proposition $20$) ProofLet $ABC$ be a triangle We can extend $BA$ past $A$ into a straight line. There exists a point $D$ such that $DA = CA$. Therefore, from Isosceles Triangle has Two Equal Angles: $\angle ADC = \angle ACD$Thus by Euclid's fifth common notion: $\angle BCD > \angle BDC$Since $\triangle DCB$ is a triangle having $\angle BCD$ greater than $\angle BDC$, this means that $BD > BC$. But: $BD = BA + AD$and: $AD = AC$Thus: $BA + AC > BC$
$\blacksquare$ Historical NoteThis proof is Proposition $20$ of Book $\text{I}$ of Euclid's
The Elements. Sources
Uh-Oh! That’s all you get for now. We would love to personalise your learning journey. Sign Up to explore more. Sign Up or Login Skip for now Uh-Oh! That’s all you get for now. We would love to personalise your learning journey. Sign Up to explore more. Sign Up or Login Skip for now No worries! We‘ve got your back. Try BYJU‘S free classes today! Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses Solution The correct option is B FalseTriangle in-equality:Given that the sum of two sides of a triangle is greater than or equal to the third side.We have to determine if the given statement is true or false.We know that the sum of any two sides of a triangle is always greater than the third side.Therefore the given statement is false.Solve Textbooks Question Papers Is the sum of two sides of a triangle is greater than the third side?Hence, sum of two sides of a triangle is always greater than the third side.
Is difference of any two sides of a triangle is greater than the third side?We know that the definition of a triangle as the polygon having three sides such that the sum of any two sides is greater than the third side. Therefore, we can say that the difference between two sides is less than the third side.
Why must the sum of two sides of a triangle be greater than the third side?If the sum of the two sides is equal to the third side, then the two sides will coincide with the third side so a triangle cannot be formed. Hence, the sum of the two sides must be greater than the third side for the triangle to be formed.
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