Find the range of values for the third side of a triangle if two of its sides measure 42.7 mm and 38.03 mm.(1 point)

To find the range of values for the third side of a triangle when given two sides \( a \) and \( b \), we can use the triangle inequality theorem. The theorem states:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Where \( c \) is the length of the third side. Let's denote the two given sides as follows:

- \( a = 42.7 \, \text{mm} \)
- \( b = 38.03 \, \text{mm} \)

Using the triangle inequalities:

1. From \( a + b > c \):
\[
42.7 + 38.03 > c \Rightarrow 80.73 > c \Rightarrow c < 80.73
\]

2. From \( a + c > b \):
\[
42.7 + c > 38.03 \Rightarrow c > 38.03 - 42.7 \Rightarrow c > -4.67
\]
Since the length of a side cannot be negative, we have \( c > 0 \).

3. From \( b + c > a \):
\[
38.03 + c > 42.7 \Rightarrow c > 42.7 - 38.03 \Rightarrow c > 4.67
\]

Combining these inequalities, we find:
\[
4.67 < c < 80.73
\]

Thus, the range of values for the third side \( c \) is:
\[
4.67 < c < 80.73
\]

Therefore, the correct response is:
**4.67 < x < 80.73**.