To determine which set is a subset of [tex]\( I \)[/tex], we need to analyze the definitions and properties of each type of triangle in relation to an isosceles triangle.
1. Set [tex]\( E \)[/tex] (Equilateral Triangles):
- An equilateral triangle is a triangle in which all three sides are of equal length.
- Every equilateral triangle is also isosceles (because it has at least two equal sides).
- Therefore, [tex]\( E \subseteq I \)[/tex].
2. Set [tex]\( S \)[/tex] (Scalene Triangles):
- A scalene triangle is a triangle in which all three sides are of different lengths.
- Since none of the sides are equal, a scalene triangle can never be isosceles.
- Therefore, [tex]\( S \nsubseteq I \)[/tex].
3. Set [tex]\( A \)[/tex] (Acute Triangles):
- An acute triangle is a triangle in which all three angles are less than 90 degrees.
- Acute triangles can be scalene, isosceles, or equilateral, so not all acute triangles are isosceles.
- Therefore, [tex]\( A \nsubseteq I \)[/tex].
4. Set [tex]\( R \)[/tex] (Right Triangles):
- A right triangle is a triangle in which one of the angles is exactly 90 degrees.
- Right triangles can be scalene or isosceles, so not all right triangles are isosceles.
- Therefore, [tex]\( R \nsubseteq I \)[/tex].
Based on the above analysis, the set that is a subset of [tex]\( I \)[/tex] (the set of isosceles triangles) is [tex]\( E \)[/tex] (the set of equilateral triangles).
Thus, the subset of [tex]\( I \)[/tex] is:
[tex]\[ \boxed{E} \][/tex]
