The range for the measure of the third side \( c \) is \( -6 < c < 16 \). The length must be greater than 6 and less than 16 to satisfy the triangle inequality theorem.
To find the range for the measure of the third side of a triangle with side lengths 5 and 11, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let \( a \), \( b \), and \( c \) be the side lengths of a triangle. The triangle inequality theorem can be expressed as:
\[ a + b > c \]
\[ a + c > b \]
\[ b + c > a \]
In this case, let \( a = 5 \) and \( b = 11 \). We want to find the range for \( c \), the measure of the third side. Apply the inequalities:
\[ 5 + 11 > c \implies 16 > c \]
\[ 5 + c > 11 \implies c > 6 \]
\[ 11 + c > 5 \implies c > -6 \]
So, the range for the measure of the third side \( c \) is \( -6 < c < 16 \). The length must be greater than 6 and less than 16 to satisfy the triangle inequality theorem.