UPSC NDA Math Model Paper 8

Given: R={(T1,T2):T1. is similar to T2 where T1,T2∈T} and T is the set of all triangles in a plane
Reflexive:
As we know that, every triangle is similar to itself, so (T1,T1)∈R‌∀T1∈T
Hence, relation R is reflexive.
Symmetric:
Suppose if (T1,T2)∈R⇒T1 is similar to T2⇒T2 is also similar to T1⇒(T2,T1)∈R.
Hence, relation R is symmetric.
Transitive:
Now suppose, (T1,T2),(T2,T3)∈R⇒T1 is similar to T2 and T2 is similar to T3⇒T1 is similar to T3 . So (T1,T3)∈R.
Hence, relation R is transitive.
Hence, relation R is an equivalence relation.