Question

Prove that the Sum of any two sides of a triangle is greater than the third side. Triangle Inequality Theorem.

Answer 1

Step:proof

Firstly by Construction: In$\Delta ABC$, extend $AB$ to $D$ in such a way that $AD=AC.$

In $\Delta DBC$, as the angles opposite to equal sides are always equal, so,

$\angle ADC=\angle ACD$

Therefore,$\angle BCD>\angle BDC$

As the sides opposite to the greater angle is longer, so,

$$\begin{gathered} BD>BC \\ AB+AD>BC \end{gathered}$$

Since $AD=AC$, then,

$AB+AC>BC$

Hence, sum of two sides of a triangle is always greater than the third side.