Consider the following sets:

[tex]\[ U = \{ \text{all triangles} \} \][/tex]
[tex]\[ E = \{ x \mid x \in U \text{ and } x \text{ is equilateral} \} \][/tex]
[tex]\[ I = \{ x \mid x \in U \text{ and } x \text{ is isosceles} \} \][/tex]
[tex]\[ S = \{ x \mid x \in U \text{ and } x \text{ is scalene} \} \][/tex]
[tex]\[ A = \{ x \mid x \in U \text{ and } x \text{ is acute} \} \][/tex]
[tex]\[ O = \{ x \mid x \in U \text{ and } x \text{ is obtuse} \} \][/tex]
[tex]\[ R = \{ x \mid x \in U \text{ and } x \text{ is right} \} \][/tex]

Which is a subset of \( I \)?

A. \( E \)
B. \( S \)
C. \( A \)
D. [tex]\( R \)[/tex]



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]