Answer :
Let's analyze the definitions and relationships between the sets of triangles given:
1. [tex]\( E \)[/tex] is the set of equilateral triangles.
2. [tex]\( I \)[/tex] is the set of isosceles triangles.
3. [tex]\( S \)[/tex] is the set of scalene triangles.
4. [tex]\( A \)[/tex] is the set of acute triangles.
5. [tex]\( O \)[/tex] is the set of obtuse triangles.
6. [tex]\( R \)[/tex] is the set of right triangles.
To determine which of these sets is a subset of [tex]\( I \)[/tex], let's consider the properties of each set relative to [tex]\( I \)[/tex]:
1. Equilateral triangles ([tex]\( E \)[/tex]): By definition, an equilateral triangle has all three sides equal. Since equilateral triangles fit the definition of isosceles triangles (which have at least two equal sides), every equilateral triangle is also an isosceles triangle. Hence, [tex]\( E \subseteq I \)[/tex].
2. Scalene triangles ([tex]\( S \)[/tex]): A scalene triangle has all sides of different lengths, meaning no two sides are equal. This does not satisfy the condition for being isosceles. Thus, [tex]\( S \)[/tex] is not a subset of [tex]\( I \)[/tex].
3. Acute triangles ([tex]\( A \)[/tex]): An acute triangle has all internal angles less than [tex]\( 90^\circ \)[/tex]. This property is unrelated to the lengths of the sides, so acute triangles can be equilateral, isosceles, or scalene. Therefore, not all acute triangles are isosceles, and [tex]\( A \)[/tex] is not a subset of [tex]\( I \)[/tex].
4. Obtuse triangles ([tex]\( O \)[/tex]): An obtuse triangle has one internal angle greater than [tex]\( 90^\circ \)[/tex]. Similar to acute triangles, the sides can vary in length, and obtuse triangles can be scalene or isosceles. Thus, [tex]\( O \)[/tex] is not a subset of [tex]\( I \)[/tex].
5. Right triangles ([tex]\( R \)[/tex]): A right triangle has one internal angle equal to [tex]\( 90^\circ \)[/tex]. The side lengths of right triangles can also vary, and they can be isosceles or scalene. So, [tex]\( R \)[/tex] is not necessarily a subset of [tex]\( I \)[/tex] either.
Given our analysis, the set which is definitively a subset of [tex]\( I \)[/tex] is [tex]\( E \)[/tex].
Hence, the correct subset of [tex]\( I \)[/tex] from the given options is:
[tex]\[ \boxed{E} \][/tex]
1. [tex]\( E \)[/tex] is the set of equilateral triangles.
2. [tex]\( I \)[/tex] is the set of isosceles triangles.
3. [tex]\( S \)[/tex] is the set of scalene triangles.
4. [tex]\( A \)[/tex] is the set of acute triangles.
5. [tex]\( O \)[/tex] is the set of obtuse triangles.
6. [tex]\( R \)[/tex] is the set of right triangles.
To determine which of these sets is a subset of [tex]\( I \)[/tex], let's consider the properties of each set relative to [tex]\( I \)[/tex]:
1. Equilateral triangles ([tex]\( E \)[/tex]): By definition, an equilateral triangle has all three sides equal. Since equilateral triangles fit the definition of isosceles triangles (which have at least two equal sides), every equilateral triangle is also an isosceles triangle. Hence, [tex]\( E \subseteq I \)[/tex].
2. Scalene triangles ([tex]\( S \)[/tex]): A scalene triangle has all sides of different lengths, meaning no two sides are equal. This does not satisfy the condition for being isosceles. Thus, [tex]\( S \)[/tex] is not a subset of [tex]\( I \)[/tex].
3. Acute triangles ([tex]\( A \)[/tex]): An acute triangle has all internal angles less than [tex]\( 90^\circ \)[/tex]. This property is unrelated to the lengths of the sides, so acute triangles can be equilateral, isosceles, or scalene. Therefore, not all acute triangles are isosceles, and [tex]\( A \)[/tex] is not a subset of [tex]\( I \)[/tex].
4. Obtuse triangles ([tex]\( O \)[/tex]): An obtuse triangle has one internal angle greater than [tex]\( 90^\circ \)[/tex]. Similar to acute triangles, the sides can vary in length, and obtuse triangles can be scalene or isosceles. Thus, [tex]\( O \)[/tex] is not a subset of [tex]\( I \)[/tex].
5. Right triangles ([tex]\( R \)[/tex]): A right triangle has one internal angle equal to [tex]\( 90^\circ \)[/tex]. The side lengths of right triangles can also vary, and they can be isosceles or scalene. So, [tex]\( R \)[/tex] is not necessarily a subset of [tex]\( I \)[/tex] either.
Given our analysis, the set which is definitively a subset of [tex]\( I \)[/tex] is [tex]\( E \)[/tex].
Hence, the correct subset of [tex]\( I \)[/tex] from the given options is:
[tex]\[ \boxed{E} \][/tex]
