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 each of the types of triangles mentioned:
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Right Isosceles: A right isosceles triangle has one right angle and two sides of equal length. This is possible and adheres to the Triangle Inequality Theorem.
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Obtuse Equilateral: An equilateral triangle has all sides equal and all angles equal to 60 degrees. It cannot be obtuse, as obtuse triangles have one angle greater than 90 degrees. Therefore, an obtuse equilateral triangle is not possible.
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Equilateral Scalene: An equilateral triangle by definition has three equal sides. A scalene triangle has all sides of different lengths. So, an equilateral scalene triangle is not possible.
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Right Equilateral: Similar to the earlier argument, an equilateral triangle cannot have a right angle because all angles in an equilateral triangle are 60 degrees. Thus, a right equilateral triangle is not possible.
Based on this analysis, the only type of triangle that is possible according to the Triangle Inequality Theorem is:
Right Isosceles.