To determine the possible values for the length of the third side of a triangle, we apply the triangle inequality theorem, which states:

For any triangle with sides $a$, $b$, and $c$:

$a + b > c$ $a + c > b$ $b + c > a$

Given $a = 10$ and $b = 5$, let $c$ be the length of the third side. Applying the inequalities:

1. $10 + 5 > c$
   $c < 15$

2. $10 + c > 5$
   $c > -5$ (which is always true for positive lengths)

3. $5 + c > 10$
   $c > 5$

Combining the valid conditions, we get:

$5 < c < 15$

Thus, the possible values for the length of the third side $c$ lie in the interval:

$(5, 15)$

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Related Questions:

1. How does the triangle inequality theorem apply to different types of triangles?
2. What happens if the given sides are equal (e.g., 10 and 10)?
3. Can a triangle exist if the sum of two sides equals the third side?
4. How would the interval change if one given side was shorter (e.g., 4 instead of 5)?
5. What are the possible angles in a triangle with sides 10, 5, and a third side within this range?

Tip:

The sum of the two shortest sides must always be greater than the longest side for a valid triangle!