Question:medium

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

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A relation is an equivalence relation if it is reflexive, symmetric, and transitive.
Updated On: Nov 26, 2025
  • Reflexive only
  • Symmetric only
  • Symmetric and reflexive only
  • Equivalence relation
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The Correct Option is D

Solution and Explanation

Step 1: Reflexivity Check
A relation is reflexive if all elements relate to themselves. Here, every triangle is congruent to itself: \( \triangle A \cong \triangle A \). Therefore, relation R is reflexive.
Step 2: Symmetry Check
A relation is symmetric if \( a \) relates to \( b \) implies \( b \) relates to \( a \). If \( \triangle A \cong \triangle B \), then \( \triangle B \cong \triangle A \). Thus, relation R is symmetric. 
Step 3: Transitivity Check
A relation is transitive if \( a \) relates to \( b \) and \( b \) relates to \( c \) implies \( a \) relates to \( c \). If \( \triangle A \cong \triangle B \) and \( \triangle B \cong \triangle C \), then \( \triangle A \cong \triangle C \). Thus, relation R is transitive. 
Step 4: Conclusion
Relation R is reflexive, symmetric, and transitive, confirming it is an equivalence relation. Answer is (D). 
 

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