Expert Answer to "Prove that the relation of congruence is an..."

High School
Geometry
High school geometry
Congruence

illusiia

Answered question

2020-11-08

Prove that the relation of congruence is an equivalence relation.

Answer & Explanation

ensojadasH

Skilled2020-11-09Added 100 answers

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:

S i n c e a \equiv a \forall a, t h e r e l a t i o n \equiv i s r e f l e x i v e

.
Symmetry:

S i n c e a \equiv b t h e n b \equiv a, t h e r e l a t i o n \equiv i s s y m m e t r y

.
Transitivity:

S i n c e a \equiv b a n d b \equiv c t h e n a \equiv c, t h e r e l a t i o n \equiv i s t r a n s i t i v e

.
Hence, the relation of congruence is an equivalence relation.

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