The Triangle Inequality 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.
Now, let's analyze the options given:
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Right equilateral: This triangle is not possible because an equilateral triangle has all three sides equal and all angles equal to 60 degrees. A right triangle must have one angle equal to 90 degrees, which cannot be satisfied in an equilateral triangle.
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Obtuse equilateral: Similar to the right triangle case, an equilateral triangle cannot have an angle greater than 60 degrees. Therefore, it's impossible for a triangle to be both obtuse and equilateral.
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Equilateral scalene: This is contradictory because an equilateral triangle has all sides equal, while a scalene triangle has no sides equal.
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Right isosceles: This triangle is possible. In a right isosceles triangle, there are two equal sides, and the angle between them is 90 degrees. For instance, if the two equal sides are each of length \( x \), the hypotenuse would be \( x\sqrt{2} \), which satisfies the triangle inequality.
Therefore, the only possible type of triangle based on the Triangle Inequality Theorem from the provided options is right isosceles.