4 The sum of the lenghts of any two sides.

of a

triangle is always greater than the

lent lenght of third side .formula

# 4 The sum of the lenghts of any two sides.

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Answer:Note: This rule must be satisfied for all 3 conditions of the sides.

Step-by-step explanation:Consider three line segments with lengths a , b , and c.

Construct two circles with radii a and b at the end points of the segment with length c .

There are three possible relationships between a+b , and c :

a+b<c: the circles will not intersect. There will be a gap on segment c between the two circles.

a+b=c: the circles will be tangent to each other. They will touch each other at a point on c .

a+b>c: the circles might

intersect each other, or

one of the circles might be large enough to fully contain the other.

That is the case when either a≥b+c , or b≥a+c.

It seems that only in the third case of a+b>c do we end up with the possibility for a triangle, and only if we are also able to apply the same reasoning to the other two sides when being considered as the base.

So, if three line segments with lengths a , b , and c can form a triangle, it must be the case that:

a+b>c ,

b+c>a , and

c+a>b .

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