Sec. 10.10 Angles between lines 175
10.10.8 Group properties of duo-angles; sensed duo-angles .........
We note the following properties of addition of duo-angles:-
(i)Given any duo-anglesαd,βdinDA(F),thesumαd+βdis a unique object
γdand it lies inDA(F).
(ii)Addition of duo-angles is commutative, that is
αd+βd=βd+αd,for allαd,βd∈DA(F).
(iii)Addition of duo-angles is associative onDA(F).
(iv)The null angle (^0) dFis a neutral element for+onDA(F).
(v)Eachαd∈DA(F)has an additive inverse inDA(F).
Proof.
(i) This is evident from the definition.
(ii) This is evident as the definition is symmetrical in the roles of the two duo-
angles.
(iii) This was established in 10.10.7.
(iv) Forαd+ (^0) dF=αd,forallαd∈DA(F).
(v) With the notation of 10.10.5 letZ 5 =sOJ(Z 4 )so thatZ 5 ≡(−x 4 ,y 4 ),andletδd
be the duo-angle inDA(F)with armsOI,OZ 5. Then, straightforwardly,αd+δd=
(^0) dF. Thus this duo-angleδdis an additive inverse forαdinDA(F). We denote it
by−αd.
These properties show that we have a commutative group. We note that
sin(−αd)=
y 4
k
=sinαd, cos(−αd)=−
x 4
k
=−cosαd,
tan(−αd)=−
y 4
x 4
=−tanαd.
Ifα=∠FQOZ 4 is a wedge-angle inA(F), withZ 4 ≡(x 4 ,y 4 )andy 4 >0, we
recall that−α=∠FQOZ 6 whereZ 6 =sOI(Z 4 )≡(x 4 ,−y 4 ).Ifαdis the duo-angle
inDA(F)with side-lines(OQ,OZ 4 )then−αdis the duo-angle inDA(F)with
side-lines(OQ,OZ 5 )whereZ 5 =sOJ(Z 4 )≡(−x 4 ,y 4 ). This inverse angle and inverse
duo-angle are linked in thatOZ 5 =OZ 6 and so|−α|◦=|−αd|◦+180.
We defineβd−αd=βd+(−αd), and this is the duo-angle in standard position
with side-linesOQandOZ 7 ,whereZ 7 ≡(x 7 ,y 7 )is the point where the line through
Qand parallel toZ 5 sOJ(Z 4 )meets the circleC(O;k)again. We callβd−αdthedif-
ference of two duo-anglesandalsoasensed duo-anglewith side-linesOZ 4 ,OZ 5
and denote it byF(OZ 4 ,OZ 5 ).IfF′is any frame of reference obtained fromFby
translation, we also define
F′(OZ 4 ,OZ 5 )=F(OZ 4 ,OZ 5 ).
Earlier names for this were a ‘complete angle’ and a ‘cross’; see Forder [7] for ap-
plications and exercises, and Forder [6, pages 120–121, 151–154] for applications,