Let T be the set of all triangles in a plane with R a relation in T given by `R={(T_1,T_2): T_1( i s c o n g r u e n t t o T)_2}`. Show that R is an equivalence relation.

Explanation

Since a relation `R` in `T` is said to be an equivalence relation if `R` is reflexive, symmetric and transitive.
(i) Since every triangle is congruent to itself
`:.R` is reflexive
(ii) `(T_(1),T_(2))epsilonRimpliesT_(1)` is congruent to `T_(2)impliesT_(2)` is congruent to `T_(1)implies(T_(2),T_(1))epsilonR`
Hence `R` is symmetric.
(iii) Let `(T_(1),T_(2))epsilonR ` and `(T_(2),T_(3))epsilonimpliesT_(1)` is congruent to `T_(2)` and `T_(2)` is congruent to `T_(3)`
`implies T_(1)` is congruent to `T_(3)implies(T_(1),T_(3))epsilonR`
`:.R` is transitive
Hence `R` is an equivalence relation.