Question

the sum of two side of a triangle is always less than the third side​

Answer 1

Proof

Triangle Inequality.png

Let 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:

∠ADC=∠ACD

Thus by Euclid's fifth common notion:

∠BCD>∠BDC

Since △DCB is a triangle having ∠BCD greater than ∠BDC, this means that BD>BC.

But:

BD=BA+AD

and:

AD=AC

Thus:

BA+AC>BC

A similar argument shows that AC+BC>BA and BA+BC>AC.