# Prove that the relation of congruence is an equivalence relation.

Question
Congruence
Prove that the relation of congruence is an equivalence relation.

2020-11-09
Step 1
The definition of a congruence depends on the type of algebraic structure under consideration.
So, for the question “Prove that the relation of congruence is an equivalence relation.” we are taking in general.
Let $$a \equiv b$$ is denoted as a is congruent to b.
Step 2
Now,
Reflexive:
$$Since\ a\ \equiv a\forall a,\ the\ relation\ \equiv is\ reflexive$$.
Symmetry:
$$Since\ a \equiv b\ then\ b \equiv\ a, the\ relation\ \equiv\ is\ symmetry$$.
Transitivity:
$$Since\ a \equiv b\ and\ b \equiv c\ then\ a \equiv c\ ,\ the\ relation\ \equiv is\ transitive$$.
Hence, the relation of congruence is an equivalence relation.

### Relevant Questions

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decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem.
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Prove directly from the definition of congruence modulo n that if a,c, and n are integers,n >1, and $$\displaystyle{a}\equiv{c}{\left(\text{mod}{n}\right)},{t}{h}{e}{n}{a}^{{3}}\equiv{c}^{{3}}{\left(\text{mod}{n}\right)}$$.
Prove that the similarity of polygons is an equivalence relation.
State the third congruence required to prove the congruence of triangles using the indicated postulate.

a)$$\displaystyle\overline{{{O}{M}}}\stackrel{\sim}{=}\overline{{{T}{S}}}$$
b)$$\displaystyle\angle{M}\stackrel{\sim}{=}\angle{S}$$
c)$$\displaystyle\overline{{{O}{N}}}\stackrel{\sim}{=}\overline{{{T}{R}}}$$
d)$$\displaystyle\angle{O}\stackrel{\sim}{=}\angle{T}$$
State the third congruence required to prove the congruence of triangles using the indicated postulate.

a)$$\displaystyle\overline{{{Z}{Y}}}\stackrel{\sim}{=}\overline{{{J}{L}}}$$
b)$$\displaystyle\angle{X}\stackrel{\sim}{=}\angle{K}$$
c)$$\displaystyle\overline{{{K}{L}}}\stackrel{\sim}{=}\overline{{{X}{Z}}}$$
d)$$\displaystyle\angle{Y}\stackrel{\sim}{=}\angle{L}$$
With of the following triangle congruence shortcuts could be used to prove PRQ = TRS

Given Data,
$$\displaystyle\angle{Q}\stackrel{\sim}{=}\angle{S}$$
$$\displaystyle\overline{{{Q}{R}}}\stackrel{\sim}{=}\overline{{{S}{R}}}$$
a)Side-Side-Side Postulate (SSS)
b)Side-Angle-Side Postulate (SAS)
c)Angle-Side-Angle Postulate (ASA)
d)Angle-Angle-Side Theorem (AAS)
A)$$\displaystyle\angle{B}{A}{C}\stackrel{\sim}{=}\angle{D}{A}{C}$$
B)$$\displaystyle\overline{{{A}{C}}}\stackrel{\sim}{=}\overline{{{B}{D}}}$$
C)$$\displaystyle\angle{B}{C}{A}\stackrel{\sim}{=}\angle{D}{C}{A}$$
D)$$\displaystyle\overline{{{A}{C}}}\stackrel{\sim}{=}\overline{{{B}{D}}}$$