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------------------------------------------------ A carpenter is searching through scraps of wood to complete the frame of a triangular shadow box.

He has two pieces of wood that measure 38.1 centimeters and 40.5 centimeters. What is the range for the measure of the third side?

[tex]
\[
\boxed{\phantom{0}} \ < \ x \ < \ \boxed{\phantom{0}}
\]
[/tex]

To find the range for the measure of the third side of a triangle, 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.

Given two sides of a triangle:
- Side A = 38.1 cm
- Side B = 40.5 cm

Let's determine the possible length for the third side, which we'll call x.

1. The first condition of the theorem is that the sum of side A and side B must be greater than x:
[tex]\[
x < 38.1 + 40.5
\][/tex]
This simplifies to:
[tex]\[
x < 78.6
\][/tex]

2. The second condition is that the sum of x and side A must be greater than side B:
[tex]\[
x + 38.1 > 40.5
\][/tex]
Solving for x, we get:
[tex]\[
x > 40.5 - 38.1
\][/tex]
Which simplifies to:
[tex]\[
x > 2.4
\][/tex]

3. The third condition is that the sum of x and side B must be greater than side A:
[tex]\[
x + 40.5 > 38.1
\][/tex]
Solving for x, we get:
[tex]\[
x > 38.1 - 40.5
\][/tex]
Which simplifies to:
[tex]\[
x > 2.4
\][/tex]

So, considering all the inequalities, the measurement for the third side, x, must satisfy:
[tex]\(2.4 < x < 78.6\)[/tex].

Therefore, the range for the measure of the third side is:
[tex]\(2.4 < x < 78.6\)[/tex].