A triangle has one side of length 29 units and another of length 40 units. Determine the range in which the length of the third side must lie.

A. [tex]-11 \ \textless \ x \ \textless \ 69[/tex]

B. [tex]11 \leq x \leq 69[/tex]

C. [tex]11 \ \textless \ x \ \textless \ 69[/tex]

D. [tex]-11 \leq x \leq 69[/tex]

To find the range in which the length of the third side of the triangle must lie, we use the triangle inequality theorem. This 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.

Let's denote the sides of the triangle as follows:
- Side 1 = 29 units
- Side 2 = 40 units
- Side 3 = x (this is the side we are trying to find the range for)

According to the triangle inequality theorem, we have the following conditions:

1. The sum of Side 1 and Side 2 must be greater than Side 3:
[tex]\[
29 + 40 > x
\][/tex]
Simplifying, we get:
[tex]\[
69 > x
\][/tex]

2. The sum of Side 1 and Side 3 must be greater than Side 2:
[tex]\[
29 + x > 40
\][/tex]
Simplifying, we get:
[tex]\[
x > 11
\][/tex]

3. The sum of Side 2 and Side 3 must be greater than Side 1:
[tex]\[
40 + x > 29
\][/tex]
Simplifying, we get:
[tex]\[
x > -11
\][/tex]
However, since x must be positive, we disregard this condition in favor of the more restrictive [tex]\( x > 11 \)[/tex].

Combining these inequalities, we determine that the length of the third side must satisfy:
[tex]\[
11 < x < 69
\][/tex]

Therefore, the correct choice is:
C. [tex]\( 11 < x < 69 \)[/tex]