Question

Let R be the relation "is congruent to" on the set of all triangles in a plane. Is R:

• A. Reflexive only
• B. Symmetric only
• C. Symmetric and reflexive only
• D. Equivalence relation

Hint:

A relation is an equivalence relation if it is reflexive, symmetric, and transitive.

Answer 1

The Correct Option is D

Step 1: {Check for Reflexivity}
A relation is reflexive if every element is related to itself. In this case, every triangle is congruent to itself. So, \( \triangle A \cong \triangle A \). Thus, the relation R is reflexive. 
Step 2: {Check for Symmetry}
A relation is symmetric if for every \( a \) related to \( b \), \( b \) is also related to \( a \). If \( \triangle A \cong \triangle B \), then \( \triangle B \cong \triangle A \). Thus, the relation R is symmetric. 
Step 3: {Check for Transitivity}
A relation is transitive if whenever \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) is also related to \( c \). If \( \triangle A \cong \triangle B \) and \( \triangle B \cong \triangle C \), then \( \triangle A \cong \triangle C \). Thus, the relation R is transitive. 
Step 4: {Conclusion}
Since the relation R is reflexive, symmetric, and transitive, it is an equivalence relation. Therefore, the correct answer is (D). 
 

Answer 2

Step 1: Understanding the relation
We are given a relation \( R \) on the set of all triangles in a plane, where the relation is defined as "is congruent to." We are tasked with determining whether this relation is an equivalence relation.

Step 2: Definition of an equivalence relation
A relation \( R \) on a set is an equivalence relation if it satisfies the following three properties:
1. Reflexivity: Every element is related to itself. For every triangle \( T \), \( T \) is congruent to \( T \). 2. Symmetry: If \( T_1 \) is related to \( T_2 \) (i.e., \( T_1 \) is congruent to \( T_2 \)), then \( T_2 \) is related to \( T_1 \) (i.e., \( T_2 \) is congruent to \( T_1 \)). 3. Transitivity: If \( T_1 \) is congruent to \( T_2 \) and \( T_2 \) is congruent to \( T_3 \), then \( T_1 \) is congruent to \( T_3 \).
Step 3: Checking for reflexivity
A triangle is congruent to itself. Therefore, for any triangle \( T \), \( T \) is congruent to \( T \). Hence, the relation is reflexive.

Step 4: Checking for symmetry
If triangle \( T_1 \) is congruent to triangle \( T_2 \), then triangle \( T_2 \) is congruent to triangle \( T_1 \). Therefore, the relation is symmetric.

Step 5: Checking for transitivity
If triangle \( T_1 \) is congruent to triangle \( T_2 \), and triangle \( T_2 \) is congruent to triangle \( T_3 \), then triangle \( T_1 \) must be congruent to triangle \( T_3 \). Therefore, the relation is transitive.

Step 6: Conclusion
Since the relation "is congruent to" satisfies all three properties—reflexivity, symmetry, and transitivity—it is an equivalence relation.

 

Equivalence relation