Sec. 9.3 Angles in standard position 137
Modified addition + of angles has the following properties:-
(i) For allα,β∈A(F),α+βis uniquely defined and lies inA(F).(ii) For allα,β∈A(F),α+β=β+α.(iii) For allα,β,γ∈A(F),(α+β)+γ=α+(β+γ).(iv) For allα∈A(F),α+ (^0) F=α.
(v)Corresponding to eachα∈A(F),thereisaβ∈A(F)such that α+β=
(^0) F.
Proof.
(i) This is evident from the definition.
(ii) This is evident as the roles ofP 1 andP 2 are interchangeable in the definition.
(iii) We note that by the last result
cos[(α+β)+γ]=cos(α+β)cosγ−sin(α+β)sinγ
and then
cos[(α+β)+γ]=[cosαcosβ−sinαsinβ]cosγ−[sinαcosβ+cosαsinβ]sinγ,
while
cos[α+(β+γ)] =cosαcos(β+γ)−sinαsin(β+γ)
=cosα[cosβcosγ−sinβsinγ]−sinα[sinβcosγ+cosβsinγ],
and these are equal. Similarly
sin[(α+β)+γ]=sin(α+β)cosγ+cos(α+β)sinγ
=[sinαcosβ+cosαsinβ]cosγ+[cosαcosβ−sinαsinβ]sinγ,
while
sin[α+(β+γ)] =sinαcos(β+γ)+cosαsin(β+γ)
=sinα[cosβcosγ−sinαsinβ]+cosα[sinβcosγ+cosβsinγ],
and these are equal. Thus(α+β)+γandα+(β+γ)are angles inA(F)with the
same cosine and the same sine and so by 9.2.2 they are equal.
(iv) Whenβ= (^0) F, in the definition we haveP 2 =Qand thenlis the midline of
|QOP 1 and soP 3 =P 1. Thusαandα+ (^0) Fare both inA(F)and they have the same
support, so they must be equal.
(v) Given any angleα∈A(F)with support|QOP 1 ,letP 2 =sOI(P 1 )andβbe
the angle inA(F)with support|QOP 2 .Thenl=OIis the midline of|P 1 OP 2 and
so in the definitionP 3 =sl(Q)=Q. Thusα+βhas support|QOQand so it is 0F.
COMMENT. The properties just listed show that(A(F),+)is a commutative
group. Because of this the familiar properties of addition, subtraction and additive
cancellation apply to it.