To find the possible lengths for the third side of the triangle, we can use the Triangle Inequality Theorem, which states that for any triangle:

1. The sum of the lengths of any two sides must be greater than the length of the third side.
2. The difference of the lengths of any two sides must be less than the length of the third side.

Given the two sides of the triangle are $6.9$ and $8.9$, let's define the third side as $x$.

We have the following three conditions based on the Triangle Inequality Theorem:

1. $6.9 + 8.9 > x$
2. $6.9 + x > 8.9$
3. $8.9 + x > 6.9$

Now, let's solve each inequality:

1. $6.9 + 8.9 > x \Rightarrow 15.8 > x \Rightarrow x < 15.8$
2. $6.9 + x > 8.9 \Rightarrow x > 8.9 - 6.9 \Rightarrow x > 2$
3. $8.9 + x > 6.9 \Rightarrow x > 6.9 - 8.9 \Rightarrow x > -2$, but this is redundant because $x > 2$ is already a stricter condition.

So the possible values for $x$ must satisfy the compound inequality:

$2 < x < 15.8$

Thus, the third side $x$ must be greater than 2 and less than 15.8.

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Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How does the Triangle Inequality Theorem apply to non-right triangles?
2. What would happen if we used an equilateral triangle instead?
3. How do you find the area of a triangle when you know two sides and the included angle?
4. What is the law of cosines, and how does it relate to the Triangle Inequality?
5. Can a triangle exist with sides of lengths 2, 2, and 4?

Tip: Always check the Triangle Inequality conditions first when working with triangle side lengths!