Congruence Relation on Set of Triangles
📺 Video Explanation
📝 Question
Let \(T\) be the set of all triangles in the Euclidean plane.
A relation \(R\) on \(T\) is defined by:
\[ aRb \iff a \text{ is congruent to } b \]
Then, \(R\) is:
- A. reflexive but not symmetric
- B. transitive but not symmetric
- C. equivalence relation
- D. none of these
✅ Solution
Triangle congruence means same shape and same size.
To check equivalence relation:
- Reflexive
- Symmetric
- Transitive
🔹 Reflexive
Every triangle is congruent to itself.
✔ Reflexive.
🔹 Symmetric
If triangle \(a\) is congruent to triangle \(b\), then triangle \(b\) is congruent to triangle \(a\).
✔ Symmetric.
🔹 Transitive
If:
\[ a\cong b \quad \text{and} \quad b\cong c \]
Then:
\[ a\cong c \]
✔ Transitive.
🎯 Final Answer
\[ \boxed{\text{R is an equivalence relation}} \]
✔ Correct option: C
🚀 Exam Shortcut
- Congruence always satisfies all three properties
- Same size + shape is an equivalence relation
- Reflexive + symmetric + transitive = equivalence