Let $T$ be the set of all triangles with $R$ a relation in $T$ given by $R = \{(T_1, T_2) : T_1 ext{ is congruent to } T_2\}$. Show that $R$ is an equivalence relation.

Question

Let $T$ be the set of all triangles with $R$ a relation in $T$ given by $R = \{(T_1, T_2) : T_1 	ext{ is congruent to } T_2\}$. Show that $R$ is an equivalence relation.

Accepted Answer

> [!TLDR]
> Show congruence of triangles is reflexive, symmetric and transitive; all three hold, so $R$ is an equivalence relation.

**Reflexive:** Every triangle $T_1$ is congruent to itself (identical in shape and size). Hence $(T_1,T_1)\in R$ for all $T_1\in T$, so $R$ is reflexive.

**Symmetric:** Let $(T_1,T_2)\in R$. Then $T_1$ is congruent to $T_2$. Congruence works both ways, so $T_2$ is congruent to $T_1$, i.e. $(T_2,T_1)\in R$. Hence $R$ is symmetric. …

Q.Let $T$ be the set of all triangles with $R$ a relation in $T$ given by $R = \{(T_1, T_2) : T_1 \text{ is congruent to } T_2\}$ . Show that $R$ is an equivalence relation.

Concept understanding — Equivalence Relation Proof

Proving a Relation is an Equivalence Relation

The three checks

A worked template

Q.Let TTT be the set of all triangles with RRR a relation in TTT given by R={(T1,T2):T1 is congruent to T2}R = \{(T_1, T_2) : T_1 \text{ is congruent to } T_2\}R={(T1​,T2​):T1​ is congruent to T2​}. Show that RRR is an equivalence relation.

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Q.Let $T$ be the set of all triangles with $R$ a relation in $T$ given by $R = \{(T_1, T_2) : T_1 \text{ is congruent to } T_2\}$ . Show that $R$ is an equivalence relation.

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