It is well-known that the shortest distance between two points is along a straight line.  This is actually a statement about side-lengths of triangles, and is not a postulate in Euclidean geometry since it follows from other postulates:

Theorem: 

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Proof:

In any triangle either all sides are the same length, or one side is longer than at least one other side.  Call the "longest side"  and let  be the third vertex.  Then we already know that  and .  Therefore (if we add another length to ) we know that  and .  So all we have to prove is that .

Draw the perpendicular segment  to side .  Then by Corollary 1,  is the shortest distance from point  to  and  is the shortest distance from point  to , so and and therefore .  But since  is between  and , .  Therefore,  by substitution.

Example:

The lengths of two sides of a triangle are 5 and 8.  Between which two numbers must the length of the third side be?

Solution:

If the length of the third side is x, then each of the following inequalities must be true:

x + 5 > 8,  x + 8 > 5, and 5 + 8  > x.  These simplify to:

x > 3,  x > –3, and 13 > x.  The middle inequality tells us nothing, but the other two tell us x is between 3 and 13.  That is, the length of the third side must be between the sum and difference of the lengths of the other two.