congruence axioms

congruence axioms

General

Congruence Relations

. Let

A

be a set and

X

=

A

×

A

. A relation on

X

is said to be a

congruence
relation

on

X

, denoted

≅

, if the following three conditions
are satisfied:

1.

(

a

,

b

)

≅

(

b

,

a

)

, for all

a

,

b

∈

A

,

2.

if

(

a

,

a

)

≅

(

b

,

c

)

, then

b

=

c

, where

a

,

b

,

c

∈

A

,

3.

if

(

a

,

b

)

≅

(

c

,

d

)

and

(

a

,

b

)

≅

(

e

,

f

)

, then

(

c

,

d

)

≅

(

e

,

f

)

, for any

a

,

b

,

c

,

d

,

e

,

f

∈

A

.

By applying

(

b

,

a

)

≅

(

a

,

b

)

twice, we see that

≅

is

reflexive

according to the third condition. From this, it is easy to that

≅

is

symmetric

, since

(

a

,

b

)

≅

(

c

,

d

)

and

(

a

,

b

)

≅

(

a

,

b

)

imply

(

c

,

d

)

≅

(

a

,

b

)

. Finally,

≅

is

transitive

, for if

(

a

,

b

)

≅

(

c

,

d

)

and

(

c

,

d

)

≅

(

e

,

f

)

, then

(

c

,

d

)

≅

(

a

,

b

)

because

≅

is symmetric and so

(

a

,

b

)

≅

(

e

,

f

)

by the third condition. Therefore, the
congruence relation is an

equivalence relation

on pairs of elements
of

A

.

Congruence Axioms

in

Ordered Geometry

. Let

(

A

,

B

)

be an
ordered geometry with

strict betweenness relation

B

.
We say that the ordered geometry

(

A

,

B

)

satisfies

the

congruence
axioms

if

1.

there is a congruence relation

≅

on

A

×

A

;

2.

if

(

a

,

b

,

c

)

∈

B

and

(

d

,

e

,

f

)

∈

B

with

–

(

a

,

b

)

≅

(

d

,

e

)

, and

–

(

b

,

c

)

≅

(

e

,

f

)

,

then

(

a

,

c

)

≅

(

d

,

f

)

;

3.

given

(

a

,

b

)

and a ray

ρ

emanating from

p

,
there exists a unique point

q

lying on

ρ

such that

(

p

,

q

)

≅

(

a

,

b

)

;

4.

given the following:

–

three rays emanating from

p

1

such that they

intersect

with a
line

ℓ

1

at

a

1

,

b

1

,

c

1

with

(

a

1

,

b

1

,

c

1

)

∈

B

, and

–

three rays emanating from

p

2

such that they intersect with a
line

ℓ

2

at

a

2

,

b

2

,

c

2

with

(

a

2

,

b

2

,

c

2

)

∈

B

,

–

(

a

1

,

b

1

)

≅

(

a

2

,

b

2

)

and

(

b

1

,

c

1

)

≅

(

b

2

,

c

2

)

,

–

(

p

1

,

a

1

)

≅

(

p

2

,

a

2

)

and

(

p

1

,

b

1

)

≅

(

p

2

,

b

2

)

,

then

(

p

1

,

c

1

)

≅

(

p

2

,

c

2

)

;

5.

given three distinct points

a

,

b

,

c

and two distinct points

p

,

q

such that

(

a

,

b

)

≅

(

p

,

q

)

. Let

H

be a closed half plane with boundary

p

⁢

q

↔

. Then there exists a unique point

r

lying on

H

such
that

(

a

,

c

)

≅

(

p

,

r

)

and

(

b

,

c

)

≅

(

q

,

r

)

.

Congruence Relations on

line segments

,

triangles

, and
angles

. With the above five congruence axioms, one may define a
congruence relation (also denoted by

≅

by abuse of notation)
on the set

S

of closed line segments of

A

by

a

⁢

b

¯

≅

c

⁢

d

¯

iff

(

a

,

b

)

≅

(

c

,

d

)

,

where

a

⁢

b

¯

(in this entry) denotes the closed line
segment with

endpoints

a

and

b

.

It is obvious that the congruence relation defined on line segments
of

A

is an equivalence relation. Next, one defines a congruence
relation on triangles in

A

:

△

⁢

a

⁢

b

⁢

c

≅

△

⁢

p

⁢

q

⁢

r

if
their sides are

congruent

:

1.

a

⁢

b

¯

≅

p

⁢

q

¯

,

2.

b

⁢

c

¯

≅

q

⁢

r

¯

, and

3.

c

⁢

a

¯

≅

r

⁢

p

¯

.

With this definition, Axiom 5 above can be restated as: given a
triangle

△

⁢

a

⁢

b

⁢

c

, such that

a

⁢

b

¯

is congruent to
a given line segment

p

⁢

q

¯

. Then there is exactly one
point

r

on a chosen side of the line

p

⁢

q

↔

such that

△

⁢

a

⁢

b

⁢

c

≅

△

⁢

p

⁢

q

⁢

r

. Not surprisingly, the congruence
relation on triangles is also an equivalence relation.

The last major congruence relation in an ordered geometry to be
defined is on angles:

∠

⁢

a

⁢

b

⁢

c

is

congruent to

∠

⁢

p

⁢

q

⁢

r

if there are

1.

a point

a

1

on

b

⁢

a

→

,

2.

a point

c

1

on

b

⁢

c

→

,

3.

a point

p

1

on

q

⁢

p

→

, and

4.

a point

r

1

on

q

⁢

r

→

such that

△

⁢

a

1

⁢

b

⁢

c

1

≅

△

⁢

p

1

⁢

q

⁢

r

1

.

It is customary to also write

∠

⁢

a

⁢

b

⁢

c

≅

∠

⁢

p

⁢

q

⁢

r

to mean
that

∠

⁢

a

⁢

b

⁢

c

is congruent to

∠

⁢

p

⁢

q

⁢

r

. Clearly for any
points

x

∈

b

⁢

a

→

and

y

∈

b

⁢

c

→

, we have

∠

⁢

x

⁢

b

⁢

y

≅

∠

⁢

a

⁢

b

⁢

c

, so that

≅

is reflexive.

≅

is also
symmetric and transitive (as the properties are inherited from the
congruence relation on triangles). Therefore, the congruence
relation on angles also defines an equivalence relation.

References

1

D. Hilbert,

Foundations of Geometry

, Open Court Publishing Co. (1971)

2

K. Borsuk and W. Szmielew,

Foundations of Geometry

, North-Holland Publishing Co. Amsterdam (1960)

3

M. J. Greenberg,

Euclidean

and

Non-Euclidean Geometries

,

Development

and History

, W. H. Freeman and Company, San Francisco (1974)

Title

congruence axioms

Canonical name

CongruenceAxioms

Date of creation

2013-03-22 15:31:59

Last modified on

2013-03-22 15:31:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Axiom

Classification

msc 51F20

Synonym

axioms of congruence

Defines

congruence relation

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