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Congruences

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Congruence is a  the state of agreement. It meaning is taken from a Latin word  congruō  which means “I meet together, I agree”. There is a difference between congruence and similarity. Congruence, as opposed to approximation, is a relation which implies a species of equivalence.

Here we are going to discuss about Congruent Triangles. They are  basically those Similar Triangles which are same in size and angle. But as we discuss above there is a difference between congruence and similarity. These triangles can also be flipped.

Congruence Definition

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1) The quality or state of agreeing, coinciding, or being congruent

2
) A statement that two numbers or geometric figures are congruent

Congruent Triangle

In the above figure triangle ABC and triangle XYZ are congruence. The definition of congruence triangle is when two or more triangle have all three sides exactly same and all three angles are same.

Here side AB = side XY
side AC = side XZ
side CB = side ZY 

Triangle Congruence

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Two triangles are congruent, if their corresponding sides are equal in length and their corresponding angles are equal in size. If triangle ABC is congruent to triangle PQR, the relationship can be written mathematically as:

 Delta ABC $\cong$ Delta PQR

Triangle Congruence

Congruence Relation

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A congruence relation (or simply congruence) is comes under abstract algebra as an equivalence relation on an algebraic structure. These structure are the group, ring, or vector space, which is compatible with the structure. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.

Then R is a congruence relation for $∘$ iff:

$∀x_{1},x_{2},y_{1},y_{2}\in S:(x_{1}Rx_{2})\wedge (y_{1}Ry_{2})=>(x_{1}∘y_{1})R(x_{2}∘y_{2})$

Congruence Theorems

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When two triangles are congruent, there are 6 facts that are true about the triangles:

the triangles have 3 sets of congruent (of equal length) sides and 
the triangles have 3 sets of congruent (of equal measure) angles.

There are several types of congruent triangles theorems

(1)
SAS (side angle side)

(2)
SSS (side side side)

(3)
ASA (angle side angle)

(4)
AAS (angle angle side)

(5)
RHS or HL (right hypotenuse side)

(6)
AAA (angle angle angle)

Congruence Theorems

AAA Statement: If in two triangles, the corresponding angles are equal, i.e., if the two triangles are equiangular, then the triangles are similar.

SSS Statement: Two triangles are congruent, if three sides of one triangle are equal to the corresponding three sides of the other triangle.

ASA Statement: If in two triangles, two angles and one side are equal then triangles are congruent. 

RHS Statement: If hypotenuse of a right angle triangle is congruent to hypotenuse of another right angle triangle and leg (perpendicular) of one right angle triangle is congruent to leg of other right angle triangle then both triangle will be congruent to each other, this is called hypotenuse leg theorem.

Congruence Symbol

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Congruence (symbol: $\cong $) is the state achieved by coming together, the state of agreement. The Latin congruō meaning “I meet together, I agree”. As an abstract term, congruence means similarity between objects. Congruence, as opposed to approximation, is a relation which implies a species of equivalence.

Congruence and Similarity

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Two shapes are said to be congruent if they are the same shape and size: that is, the corresponding sides of both shapes are the same length and corresponding angles are the same. The two triangles shown here are congruent. Shapes which are of different sizes but which have the same shape are said to be similar.
Congruence: The shapes are Congruence, if one shape can become another using Turns, Flips and/or Slides.

Similarity:  A similarity transforms a object into a similar object but the size may varies. 
The only difference between congruence and similarity is size variation. When two triangles are congruence the size will be same and in similarity size can be change.