To determine the possible values of [tex]\( n \)[/tex] for a triangle with sides measuring [tex]\( 20 \)[/tex] cm, [tex]\( 5 \)[/tex] cm, and [tex]\( n \)[/tex] cm, we need to use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Let’s apply this theorem to our problem step-by-step:
1. First Inequality:
[tex]\[
a + b > n \implies 20 + 5 > n \implies 25 > n \implies n < 25
\][/tex]
2. Second Inequality:
[tex]\[
a + n > b \implies 20 + n > 5 \implies n > 5 - 20
\][/tex]
Since the value of [tex]\( n \)[/tex] must be positive (because it represents a side length), we actually have:
[tex]\[
20 + n > 5 \implies n > 5 - 20 \implies n > -15
\][/tex]
However, [tex]\( n \)[/tex] being a positive side length means:
[tex]\[
n > 0
\][/tex]
But we only need the relevant comparison from:
[tex]\[
20 + n > 5 \implies n > -15 \text{ (this is always true for positive \( n \))}
\][/tex]
3. Third Inequality:
[tex]\[
b + n > a \implies 5 + n > 20 \implies n > 20 - 5 \implies n > 15
\][/tex]
Combining these inequalities, we get:
[tex]\[
15 < n < 25
\][/tex]
Thus, the valid range for [tex]\( n \)[/tex] that satisfies all the conditions of the triangle inequality theorem is:
[tex]\[
15 < n < 25
\][/tex]
The correct choice from the given options is:
[tex]\[
\boxed{15 < n < 25}
\][/tex]
