The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Now, let's analyze the options provided:
A. Right Isosceles: A right isosceles triangle has two sides that are equal in length and a right angle. For example, if the two equal sides are of length \( a \), then the hypotenuse will be \( \sqrt{2}a \). The sides satisfy the triangle inequality, so this type is possible.
B. Equilateral Scalene: An equilateral triangle has all three sides equal. A scalene triangle has all sides unequal. This option is contradictory because you cannot have an equilateral scalene triangle. Thus, this type is not possible.
C. Obtuse Equilateral: An equilateral triangle cannot be obtuse as it has all angles equal to \( 60^\circ \). Therefore, this type is not possible.
D. Right Equilateral: Similar to the obtuse case, an equilateral triangle cannot have a right angle, as it cannot have angles of \( 90^\circ \) when all three angles must equal \( 180^\circ \). Thus, this type is also not possible.
Based on this analysis, the only possible type of triangle is A: right isosceles.