Let's examine each statement individually to determine their validity for a triangle with side lengths [tex]\( m \)[/tex], [tex]\( n \)[/tex], and [tex]\( o \)[/tex].
1. Statement 1: [tex]\( m + o > n \)[/tex]
In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Therefore, for a triangle with sides [tex]\( m \)[/tex], [tex]\( n \)[/tex], and [tex]\( o \)[/tex], this inequality must hold true:
[tex]\[
m + o > n
\][/tex]
This follows directly from the triangle inequality theorem.
2. Statement 2: [tex]\( m + n < 0 \)[/tex]
This statement suggests that the sum of two side lengths is less than zero. However, the length of each side of a triangle is a positive number. Therefore, the sum of two positive numbers cannot be less than zero. Hence, this statement is false.
3. Statement 3: [tex]\( m - n > 0 \)[/tex]
This statement can be rewritten as:
[tex]\[
m > n
\][/tex]
This would mean that side [tex]\( m \)[/tex] is longer than side [tex]\( n \)[/tex]. While it's possible for this to be true in a specific instance, it is not necessarily true for all triangles in general. Therefore, this statement is not universally true for all triangles.
4. Statement 4: [tex]\( o - n > m \)[/tex]
This statement can be rewritten as:
[tex]\[
o > n + m
\][/tex]
According to the triangle inequality theorem, the sum of any two sides must be greater than the third side, meaning:
[tex]\[
o < n + m
\][/tex]
Therefore, this statement contradicts the triangle inequality theorem and is false.
Thus, the only statement that is always valid for any triangle with side lengths [tex]\( m \)[/tex], [tex]\( n \)[/tex], and [tex]\( o \)[/tex] is:
[tex]\[
\boxed{m + o > n}
\][/tex]
