Triangle Inequality

Theorem: Sum of the lengths of any two sides of a triangle is always greater than the third side.

Prerequisites:
Straight line (definition)

Proof:

The given figure presents two paths from A to B, i.e., a direct straight line path from A to B, and a path via C. Lengths of these two paths are $c$ and $a+b$, respectively.

Since by definition of a straight line, straight line joins the two points uniquely through the shortest path,

$\therefore\quad\;\; c < a + b$

Similar inequalities holds for the other two sides.

Hence the result.


Recommended:
Angle Sum Property of triangle
Angle Sum Property of polygon
Unique SSS Triangle