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

Question

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

Accepted Answer

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> [!TLDR]
> Congruence of triangles satisfies reflexivity, symmetry, and transitivity, so $R$ is an equivalence relation.
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> Let $T$ be the set of all triangles and $R=\{(T_1,T_2): T_1	ext{ is congruent to }T_2\}$.
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> **Reflexive:** Every triangle $T$ is congruent to itself (identical sides and angles), so $(T,T)\in R$ for all $T\in T$. Hence $R$ is reflexive.
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> **Symmetric:** If $(T_1,T_2)\in R$, then $T_1\cong T_2$. Congruence is a mutual relationship: if $T_1$ is congruent to $T_2$, then $T_2$ is congruent to $T_1$ (same sides and angles match in either direction). So $(T_2,T_1)\in R$. Hence $R$ is symmetric.
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Q.Let $T$ be the set of all triangles in a plane with $R$ , a relation in $T$ given by $R = \{(T_1, T_2) : T_1, T_2 \text{ are congruent}\}$ , 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 in a plane with RRR, a relation in TTT given by R={(T1,T2):T1,T2 are congruent}R = \{(T_1, T_2) : T_1, T_2 \text{ are congruent}\}R={(T1​,T2​):T1​,T2​ are congruent}, show that RRR is an equivalence relation.

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

Unlock everything free for 14 days