The Triangle Inequality 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, and vice versa.
Let the lengths of the two known sides be \( a = 34 \) and \( b = 51 \), and let \( c \) be the length of the third side. According to the Triangle Inequality Theorem:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Now, applying the values:
- \( 34 + 51 > c \) → \( 85 > c \) → \( c < 85 \)
- \( 34 + c > 51 \) → \( c > 51 - 34 \) → \( c > 17 \)
- \( 51 + c > 34 \) → \( c > 34 - 51 \) → This is always true since \( 34 - 51 \) is negative.
Combining the inequalities from steps 1 and 2, we find:
\[ 17 < c < 85 \]
Thus, the complete inequality indicating the possible values of the third side \( c \) is:
17 < c < 85