To determine the possible values of [tex]\( n \)[/tex] for a triangle with side lengths [tex]\( 20 \, \text{cm} \)[/tex], [tex]\( 5 \, \text{cm} \)[/tex], and [tex]\( n \, \text{cm} \)[/tex], we need to use the Triangle Inequality Theorem. The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Consider the following three inequalities derived from the theorem:
1. The sum of the first two sides must be greater than the third side:
[tex]\[
20 + 5 > n \implies 25 > n \implies n < 25
\][/tex]
2. The sum of the first and the third side must be greater than the second side:
[tex]\[
20 + n > 5 \implies n > -15
\][/tex]
Since side lengths cannot be negative, this inequality is always true and does not provide additional boundaries.
3. The sum of the second and the third side must be greater than the first side:
[tex]\[
5 + n > 20 \implies n > 15
\][/tex]
Combining these two inequalities, we have:
[tex]\[
15 < n < 25
\][/tex]
Therefore, the possible values of [tex]\( n \)[/tex] that satisfy all conditions are described by the inequality [tex]\( 15 < n < 25 \)[/tex].
So, the correct choice is:
[tex]\[ 15 < n < 25 \][/tex]
