If R is a relation on the set T of all triangles drawn in a plane defined by a R b iff a is congruent to b for all, a, b `in` T, then R is

Options

A. reflexive but not transitive
B. reflexive but not symmetric
C. symmetric but not transivtive
D. equivalence relation

Answer

To determine the properties of the relation \( R \) defined on the set \( T \) of all triangles, where \( a R b \) if and only if triangle \( a \) is congruent to triangle \( b \), we will check if \( R \) is reflexive, symmetric, and transitive. If it satisfies all three properties, then \( R \) is an equivalence relation.

### Step 1: Check Reflexivity
A relation \( R \) is reflexive if for every element \( a \) in the set \( T \), \( a R a \) holds true.

- Let \( A \) be any triangle in \( T \).
- By the definition of congruence, any triangle \( A \) is congruent to itself.
- Therefore, \( A R A \) is true.

**Conclusion:** The relation \( R \) is reflexive.

### Step 2: Check Symmetry
A relation \( R \) is symmetric if whenever \( a R b \), then \( b R a \) also holds true.

- Let \( A \) and \( B \) be any triangles in \( T \) such that \( A R B \) (i.e., \( A \) is congruent to \( B \)).
- By the properties of congruence, if triangle \( A \) is congruent to triangle \( B \), then triangle \( B \) is also congruent to triangle \( A \).
- Therefore, if \( A R B \) is true, then \( B R A \) is also true.

**Conclusion:** The relation \( R \) is symmetric.

### Step 3: Check Transitivity
A relation \( R \) is transitive if whenever \( a R b \) and \( b R c \), then \( a R c \) also holds true.

- Let \( A, B, C \) be any triangles in \( T \) such that \( A R B \) (i.e., \( A \) is congruent to \( B \)) and \( B R C \) (i.e., \( B \) is congruent to \( C \)).
- By the properties of congruence, if triangle \( A \) is congruent to triangle \( B \) and triangle \( B \) is congruent to triangle \( C \), then triangle \( A \) is also congruent to triangle \( C \).
- Therefore, \( A R C \) is true.

**Conclusion:** The relation \( R \) is transitive.

### Final Conclusion
Since the relation \( R \) is reflexive, symmetric, and transitive, it is an equivalence relation.

### Answer
Thus, \( R \) is an equivalence relation.

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