The triangle inequality theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
To determine which types of triangles are possible, we can look at specific examples. If you provide specific lengths or types (e.g., equilateral, isosceles, scalene), I can help you analyze whether those triangles satisfy the triangle inequality theorem.
For instance:
- An equilateral triangle with sides of length 3 satisfies: \(3 + 3 > 3\), \(3 + 3 > 3\), and \(3 + 3 > 3\).
- An isosceles triangle with sides 5, 5, and 7 satisfies: \(5 + 5 > 7\), \(5 + 7 > 5\), \(5 + 7 > 5\).
- A scalene triangle with sides 5, 6, and 10 does not satisfy the inequalities: \(5 + 6 = 11\) but \(5 + 10 = 15\) and does not satisfy \(5 + 6 > 10\).
Let me know the specific types or lengths of triangles you want to evaluate!
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