Answer :
To determine which set is a subset of \( I \) (the set of all isosceles triangles), let's analyze the properties of each given set:
1. Set \( E \) (Equilateral Triangles):
- An equilateral triangle is defined as a triangle with all three sides of equal length.
- For a triangle to be isosceles, it must have at least two sides of equal length.
- All equilateral triangles meet this criterion because in an equilateral triangle, all three sides are equal. Therefore, every equilateral triangle is also an isosceles triangle.
- Hence, \( E \subseteq I \).
2. Set \( S \) (Scalene Triangles):
- A scalene triangle has all sides of different lengths.
- An isosceles triangle has at least two sides of equal length.
- Because no scalene triangle can have two sides of equal length, the set of scalene triangles \( S \) cannot be a subset of \( I \).
- Hence, \( S \not\subseteq I \).
3. Set \( A \) (Acute Triangles):
- An acute triangle has all interior angles less than \(90^\circ\).
- An isosceles triangle could be acute, but it could also be right or obtuse.
- Not all acute triangles are isosceles. Acute triangles can be scalene as well.
- Therefore, \( A \) is not necessarily a subset of \( I \).
- Hence, \( A \not\subseteq I \).
4. Set \( R\) (Right Triangles):
- A right triangle has one \(90^\circ\) angle.
- A right triangle can also be isosceles if the other two sides (legs) are of equal length.
- But not all right triangles are isosceles. Right triangles can also be scalene.
- Therefore, \( R \) is not necessarily a subset of \( I \).
- Hence, \( R \not\subseteq I \).
Based on this analysis, the only correct statement is:
[tex]\[ E \subseteq I \][/tex]
Thus, the set of equilateral triangles \( E \) is a subset of \( I \) (the set of isosceles triangles). Therefore, the answer is:
[tex]\[ E \][/tex]
1. Set \( E \) (Equilateral Triangles):
- An equilateral triangle is defined as a triangle with all three sides of equal length.
- For a triangle to be isosceles, it must have at least two sides of equal length.
- All equilateral triangles meet this criterion because in an equilateral triangle, all three sides are equal. Therefore, every equilateral triangle is also an isosceles triangle.
- Hence, \( E \subseteq I \).
2. Set \( S \) (Scalene Triangles):
- A scalene triangle has all sides of different lengths.
- An isosceles triangle has at least two sides of equal length.
- Because no scalene triangle can have two sides of equal length, the set of scalene triangles \( S \) cannot be a subset of \( I \).
- Hence, \( S \not\subseteq I \).
3. Set \( A \) (Acute Triangles):
- An acute triangle has all interior angles less than \(90^\circ\).
- An isosceles triangle could be acute, but it could also be right or obtuse.
- Not all acute triangles are isosceles. Acute triangles can be scalene as well.
- Therefore, \( A \) is not necessarily a subset of \( I \).
- Hence, \( A \not\subseteq I \).
4. Set \( R\) (Right Triangles):
- A right triangle has one \(90^\circ\) angle.
- A right triangle can also be isosceles if the other two sides (legs) are of equal length.
- But not all right triangles are isosceles. Right triangles can also be scalene.
- Therefore, \( R \) is not necessarily a subset of \( I \).
- Hence, \( R \not\subseteq I \).
Based on this analysis, the only correct statement is:
[tex]\[ E \subseteq I \][/tex]
Thus, the set of equilateral triangles \( E \) is a subset of \( I \) (the set of isosceles triangles). Therefore, the answer is:
[tex]\[ E \][/tex]
