Show that the relation R defined in the set A of all triangles as $R = \{ ({T_1},\;{T_2}):{T_1}$ is similar to ${T_2}\}$ ,
is an equivalence relation. Consider three right angle triangles ${T_1}$ , with sides 3, 4, 5, ${T_2}$ with sides 5, 12, 13 and
${T_3}$ , with sides 6, 8, 10. Which triangles among ${T_1},\;{T_2}$ and ${T_3}$ are related?

$R = \{ ({T_1},\;{T_2}):{T_1}$ is similar to ${T_2}$ and ${T_1}$ , ${T_2}$ are triangles.

(i) Reflexivity
We know that each triangle is similar to itself and thus $({T_1},\;{T_1}) \in R.\;\;\;\therefore \;\;R$ is reflexive.

(ii) Symmetry
Also, two triangles are similar.
Then, ${T_1} \sim {T_2} \Rightarrow \;\;{T_2} \sim {T_1}.\;\;\;\;\therefore \;\;R$ is symmetric.

(iii) Transitivity
Again, if if ${T_1} \sim {T_2}$ and ${T_2} \sim {T_3} \Rightarrow {T_1} \sim {T_3}$ $\therefore \;\;\;R$ is transitive.
Hence, R is an equivalence relation.

We are given three right angled triangles ${T_1},\;\;{T_2}\;\;and\;\;{T_3}.$

${T_1}$ with sides 3, 4, 5 ; ${T_2}$ with sides 5, 12, 13 and ${T_3}$ with sides 6, 8, 10

We know that two triangles are similar if corresponding sides are proportional. We observe that ${T_1}$ , and ${T_3}$ are similar because $\cfrac{3}{6} = \cfrac{4}{8} = \cfrac{5}{{10}}\left( { = \cfrac{1}{2}} \right)$

Hence, triangles ${T_1}$ , and ${T_3}$ are related.