angle

angle

1

Definition

In an

ordered geometry

S

, given a point

p

let

Π

⁢

(

p

)

be the family of
all rays emanating from it. Let

α

,

β

∈

Π

⁢

(

p

)

such that

α

≠

β

and

α

≠

-

β

. The

angle

between rays (

http://

planetmath

.org/BetweennessInRays

)

α

and

β

at

p

is

{

ρ

∈

Π

⁢

(

p

)

∣

ρ

⁢

is between

⁢

α

⁢

and

⁢

β

}

.

This angle
is denoted by

∠

⁢

α

⁢

p

⁢

β

. The two rays

α

and

β

are the

sides

of the angle, and

p

the

vertex

of the
angle. Since any point (other
than the source

p

) on a ray uniquely determines the ray, we may
also write the angle by

∠

⁢

a

⁢

p

⁢

b

, whenever we have points

a

∈

α

and

b

∈

β

.

The notational device given for the angle suggests
the possibility of defining an

angle between two line

segments satisfying certain conditions:
let

p

⁢

q

¯

and

q

⁢

r

¯

be two

open line segments

with a common

endpoint

q

. The angle between the two open line segments is the angle between the
rays

q

⁢

p

→

and

q

⁢

r

→

. In this case, we may denote the angle by

∠

⁢

p

⁢

q

⁢

r

.

Suppose

ℓ

is a line and

p

a point

lying on

ℓ

. We have
two opposite rays emanating from

p

that lie on

ℓ

. Call them

σ

and

-

σ

. Any ray

ρ

emanating from a point

p

that does not lie on

ℓ

produces two angles at

p

, one between

ρ

and

σ

and the other between

ρ

and

-

σ

.
These two angles are said to be

supplement

of one another, or
that

∠

⁢

σ

⁢

p

⁢

ρ

is

supplementary

of

∠

⁢

(

-

σ

)

⁢

p

⁢

ρ

. Every angle has exactly two supplements.

2

Ordering of Angles

Let

S

be an ordered geometry and

ρ

a ray in

S

with source
point

p

. Consider the set

E

of all angles whose one side is

ρ

. Define an

ordering

on

E

by the following rule: for

∠

⁢

α

⁢

p

⁢

ρ

,

∠

⁢

β

⁢

p

⁢

ρ

∈

E

,

1.

∠

⁢

α

⁢

p

⁢

ρ

=

∠

⁢

β

⁢

p

⁢

ρ

if

α

=

β

,

2.

∠

⁢

α

⁢

p

⁢

ρ

<

∠

⁢

β

⁢

p

⁢

ρ

if

α

∈

∠

⁢

β

⁢

p

⁢

ρ

, and

3.

∠

⁢

α

⁢

p

⁢

ρ

>

∠

⁢

β

⁢

p

⁢

ρ

if

β

∈

∠

⁢

α

⁢

p

⁢

ρ

.

The

ordering relation

above is

well-defined

. However, it is quite
limited, because there is no way to compare angles if the pair (of
angles) do not share a common side. This can be remedied with an
additional set of axioms on the

geometry

: the

axioms of congruence

.

In an ordered geometry satisfying the

congruence axioms

, we have the

concept

of angle

congruence

. This

binary relation

turns out to be an

equivalence relation

, so we can form the set of

equivalence classes

on angles. Each equivalence class of angles is called a

free
angle

. Any member of a free angle

𝔞

is called a
representative of

𝔞

, which is simply an angle of form

∠

⁢

a

⁢

b

⁢

c

, where

b

is the source of two rays

b

⁢

a

→

and

b

⁢

c

→

. We write

𝔞

=

[

∠

⁢

a

⁢

b

⁢

c

]

. It is easy to
see that given a point

p

and a ray

ρ

emanating from

p

, we
can find, in each free angle, a representative whose one side is

ρ

. In other words, for any free angle

𝔞

, it is
possible to write

𝔞

=

[

∠

⁢

α

⁢

p

⁢

ρ

]

for some ray

α

.

Now we are ready to define orderings on angles in general. In fact,
this this done via free angles. Let

𝔄

be the set of
all free angles in an ordered geometry satisfying the congruence
axioms, and

𝔞

,

𝔟

∈

𝔄

. Write

𝔞

=

[

∠

⁢

α

⁢

p

⁢

ρ

]

and

𝔟

=

[

∠

⁢

β

⁢

p

⁢

ρ

]

. We say that

𝔞

<

𝔟

if ray

α

is
between

β

and

ρ

. The other inequality is dually defined.
This is a well-defined binary relation. Given the ordering on free
angles, we define

∠

⁢

α

⁢

p

⁢

β

<

∠

⁢

γ

⁢

q

⁢

δ

if

[

∠

⁢

α

⁢

p

⁢

β

]

<

[

∠

⁢

γ

⁢

q

⁢

δ

]

.

Let

ℓ

be a line and

p

a point lying on

ℓ

. The point

p

determines two opposite rays

ρ

and

-

ρ

lying on

ℓ

. Any
ray

σ

emanating from

p

that is distinct from either

ρ

and

-

ρ

determines exactly two angles:

∠

⁢

ρ

⁢

p

⁢

σ

and

∠

⁢

(

-

ρ

)

⁢

p

⁢

σ

. These two angles are said to be
supplements of one another, or that one is supplementary of the
other.

In an ordered geometry satisfying the congruence axioms,
supplementary free angles are defined if each contains a
representative that is supplementary to one another. Given two
supplementary free angles

𝔞

,

𝔟

, we may make
comparisons of the two:

•

if

𝔞

=

𝔟

, then we say that

𝔞

is a

right free angle

, or simple a

right angle

. Clearly

𝔟

is a right angle if

𝔞

is;

•

if

𝔞

>

𝔟

, then

𝔞

is called
an

obtuse free angle

, or an

obtuse angle

. The
supplement of an obtuse angle is called an

acute free angle

,
or an

acute angle

. Thus,

𝔟

is acute if

𝔞

is obtuse.

Given any two free angles, we can always compare them. In other
words, the

law of trichotomy

is satisfied by the ordering of free
angles: for any

𝔞

and

𝔟

, exactly one of

𝔞

>

𝔟

𝔞

=

𝔟

𝔞

<

𝔟

is true.

3

Operations on Angles

Let

S

be an ordered geometry satisfying the congruence axioms and

𝔞

and

𝔟

are two free angles. Write

𝔞

=

[

∠

⁢

α

⁢

p

⁢

β

]

and

𝔟

=

[

∠

⁢

β

⁢

p

⁢

γ

]

. If

β

is between

α

and

γ

, we define
an “

addition

” of

𝔞

and

𝔟

, written

𝔞

+

𝔟

as the free angle

𝔠

with
representative

∠

⁢

α

⁢

p

⁢

γ

. In symbol, this says that if

β

is between

α

and

γ

, then

[

∠

⁢

α

⁢

p

⁢

β

]

+

[

∠

⁢

β

⁢

p

⁢

γ

]

=

[

∠

⁢

α

⁢

p

⁢

γ

]

.

This is a
well-defined

binary operation

, provided that

one free angle is
between the other two

. Therefore, the sum of a pair of

supplementary angles

is not defined! In addition, if

𝔞

and

𝔠

are two free angles, such that there exists a
free angle

𝔟

with

𝔞

+

𝔟

=

𝔠

, then

𝔟

is
unique and we denote it by

𝔠

-

𝔞

. It is also
possible to define the

multiplication

of a free angle by a positive
integer, provided that the resulting angle is a well-defined free
angle. Finally,

division

of a free angle by

positive

integral powers
of 2 can also be defined.

4

Angle Measurement

An angle measure

𝒜

is a

function

defined on free angles
of an ordered geometry

S

with the congruence axioms, such that

1.

𝒜

is real-valued and positive,

2.

𝒜

is

additive

; in other words,

𝒜

⁢

(

𝔞

+

𝔟

)

=

𝒜

⁢

(

𝔞

)

+

𝒜

⁢

(

𝔟

)

, if

𝔞

+

𝔟

is defined;

Here are some

properties

:

•

if

𝒜

⁢

(

𝔞

)

=

𝒜

⁢

(

𝔟

)

, then

𝔞

=

𝔟

.

•

𝔞

>

𝔟

iff

𝒜

⁢

(

𝔞

)

>

𝒜

⁢

(

𝔟

)

.

•

for any free angle

𝔞

, denote its supplement by

𝔞

s

. Then

𝒜

⁢

(

𝔞

)

+

𝒜

⁢

(

𝔞

s

)

is a positive

constant

r

𝒜

that does not depend on

𝔞

.

•

𝒜

is bounded above by

r

𝒜

.

•

if

𝒜

and

ℬ

are angle measures, then

𝒜

+

ℬ

defined by

(

𝒜

+

ℬ

)

⁢

(

𝔞

)

=

𝒜

⁢

(

𝔞

)

+

ℬ

⁢

(

𝔞

)

is an angle
measure too.

•

if

𝒜

is an angle measure, then for any positive
real number

r

,

r

⁢

𝒜

defined by

(

r

⁢

𝒜

)

⁢

(

𝔞

)

=

r

⁢

(

𝒜

⁢

(

𝔞

)

)

is also
an angle measure. In the event that

r

is an integer such that

r

⁢

𝔞

makes sense, we also have

r

⁢

(

𝒜

⁢

(

𝔞

)

)

=

𝒜

⁢

(

r

⁢

𝔞

)

.

If

S

is a

neutral geometry

, then we impose a third requirement for
a function to be an angle measure:

3.

for any real number

r

with

0

<

r

<

r

𝒜

, there is a
free angle

𝔞

such that

𝒜

⁢

(

𝔞

)

=

r

.

Once the

measure

of a free angle is defined, one can next define the
measure of an angle: let

𝒜

be a measure of the free
angles, define

𝒜

′

on angles by

𝒜

′

⁢

(

∠

⁢

α

⁢

p

⁢

β

)

=

𝒜

⁢

(

[

∠

⁢

α

⁢

p

⁢

β

]

)

. This is a well-defined function. It is easy to
see that

𝒜

′

⁢

(

∠

⁢

α

⁢

p

⁢

β

)

=

𝒜

′

⁢

(

∠

⁢

γ

⁢

q

⁢

δ

)

iff

∠

⁢

α

⁢

p

⁢

β

≅

∠

⁢

γ

⁢

q

⁢

δ

, and

𝒜

′

⁢

(

∠

⁢

α

⁢

p

⁢

β

)

>

𝒜

′

⁢

(

∠

⁢

γ

⁢

q

⁢

δ

)

iff

∠

⁢

α

⁢

p

⁢

β

>

∠

⁢

γ

⁢

q

⁢

δ

.

Two popular angle measures are the degree measure and the radian
measure. In the degree measure,

r

𝒜

=

180

∘

. In
the radian measure,

r

𝒜

=

π

.

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

angle

Canonical

name

Angle

Date of creation

2013-03-22 15:32:36

Last modified on

2013-03-22 15:32:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

22

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51F20

Classification

msc 51G05

Synonym

supplement

Related topic

PaschsTheorem

Related topic

BetweennessInRays

Related topic

SupplementaryAngles

Defines

supplementary

Defines

right angle

Defines

between rays

Defines

crossbar theorem

Defines

free angle

Defines

acute angle

Defines

obtuse angle

Defines

angle measure

Defines

side

Defines

vertex

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