To find missing sides of a triangle, we can use something called the Triangle Inequality Theorem.

Here’s the main idea:

The lengths of any two sides must be longer than the length of the third side.

We can write this rule like this for sides labeled as (a), (b), and (c):

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

Example:

Let’s say you know two sides of a triangle:

• (a = 5)
• (b = 7)

Now, we want to find out what the third side, (c), could be.

We can use our rules to set up some inequalities:

1. From (5 + 7 > c):

   This means that (c) must be less than 12.

2. From (5 + c > 7):

   Here, we find that (c) needs to be greater than 2.

3. From (7 + c > 5):

   This will always be true because (c) will always be a positive number.

Putting all this together, we find that (c) must be between 2 and 12.

So, we can say:

[2 < c < 12]