In order to determine which set is a subset of [tex]\( I \)[/tex] (isosceles triangles), let's explore the properties of the triangles in the given sets:
- Equilateral Triangles (E): These are triangles with all three sides equal. Since they have at least two equal sides, they are also isosceles. Therefore, every equilateral triangle is an isosceles triangle, which means [tex]\( E \subseteq I \)[/tex].
- Scalene Triangles (S): These are triangles with all three sides of different lengths. Since they do not have at least two equal sides, no scalene triangle can be isosceles. Thus, [tex]\( S \nsubseteq I \)[/tex].
- Acute Triangles (A): These are triangles where all angles are less than 90 degrees. Acute triangles can be equilateral, isosceles, or scalene. Since not all acute triangles must be isosceles, [tex]\( A \nsubseteq I \)[/tex].
- Right Triangles (R): These are triangles with one angle exactly 90 degrees. Right triangles can also be isosceles (if the two legs are of equal length), but they can also be scalene. Therefore, not all right triangles are isosceles, so [tex]\( R \nsubseteq I \)[/tex].
Considering each set, the only set where all members must also be members of the isosceles triangles is [tex]\( E \)[/tex] (equilateral triangles). Therefore, the subset of [tex]\( I \)[/tex] is:
[tex]\( E \)[/tex]
