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To find the range of possible values for the third side \(x\) in a triangle, given the lengths of two sides (63 and 89), we can apply the **Triangle Inequality Theorem**. This 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.

For the sides of length \(a = 63\), \(b = 89\), and the third side \(x\), we need to satisfy these three conditions:

1. \( x + 63 > 89 \)
2. \( x + 89 > 63 \)
3. \( 63 + 89 > x \)

Let's solve these inequalities one by one:

1. \( x + 63 > 89 \) \\
   \( x > 89 - 63 \) \\
   \( x > 26 \)

2. \( x + 89 > 63 \) \\
   This inequality is always true, as any positive value of \(x\) will satisfy it.

3. \( 63 + 89 > x \) \\
   \( 152 > x \) \\
   \( x < 152 \)

Thus, the range for the third side \(x\) is:
\[
26 < x < 152
\]

So, the number that belongs in the green box is **26**.

Would you like further details on this process?

Here are 5 related questions to extend this concept:
1. How would the range change if one of the sides was smaller than 63?
2. Can the triangle inequality theorem be applied to non-Euclidean geometry?
3. How does the triangle inequality theorem relate to the Pythagorean theorem?
4. What happens to the range of the third side if the two given sides are equal?
5. How does the area of a triangle change as the third side approaches its upper or lower limit?

**Tip:** When solving for unknown sides in triangles, always check if the given side lengths can form a valid triangle using the triangle inequality theorem.

Given 63 and 89 as the lengths of two sides of a triangle, find the range of values for the third side.

Learn how to apply the Triangle Inequality Theorem to find the range of possible values for the third side of a triangle with given side lengths of 63 and 89. A detailed step-by-step solution is provided, suitable for students in Grades 8-10.

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