Geometry with Trigonometry

168 Complex coordinates; sensed angles; angles between lines Ch. 10

10.10 Anglesbetweenlines .........................

10.10.1Motivation ..............................

Since cos( (^180) F+θ)=−cosθ,sin( (^180) F+θ)=−sinθ,wehavethattan( (^180) F+
θ)=tanθ. Thus results that tanθis constant do not imply thatθ is an angle of
constant magnitude. To extract more information from such situations, we develop
new material. This also deals with the rather abrupt transitions in results such as
those in 10.9.1 and 10.9.2.

10.10.2 Duo-sectors ............................

Letl 1 ,l 2 be lines intersecting at a pointZ 1 .Whenl 1 =l 2 ,letZ 2 ,Z 3 ∈l 1 withZ 1
betweenZ 2 andZ 3 ,andletZ 4 ,Z 5 ∈l 2 withZ 1 betweenZ 4 andZ 5. Then the union

IR(|Z 2 Z 1 Z 4 )∪IR(|Z 3 Z 1 Z 5 )

we shall call aduo-sectorwith side-linesl 1 andl 2 ; we shall denote it byD 1. Similarly

IR(|Z 2 Z 1 Z 5 )∪IR(|Z 3 Z 1 Z 4 )

is also a duo-sector with side-linesl 1 andl 2 , and we shall denote it byD 2.

Z 1

Z 2

Z 4

Z 3

Z 5

D 1

D 1

l 1

l 2

Figure 10.11.

Z 1

Z 2

Z 4

Z 3

Z 5

l l^4

3

D 1

D 1

D 2

D 2

Figure 10.12.

The mid-linel 3 of|Z 2 Z 1 Z 4 is also the mid-line of|Z 3 Z 1 Z 5 and it lies entirely inD 1.
The mid-linel 4 of|Z 2 Z 1 Z 5 is also the mid-line of|Z 3 Z 1 Z 4 and it lies entirely inD 2.
We call{l 3 ,l 4 }thebisectorsof the line pair{l 1 ,l 2 }and usel 3 to identifyD 1 ,l 4 to
identifyD 2.
In Figure 10.12 suppose that|∠Z 2 Z 1 Z 4 |◦=|∠Z 3 Z 1 Z 5 |◦=xand|∠Z 4 Z 1 Z 3 |◦=y
then we have 2x+ 2 y=360 and sox+y=180. It follows that^12 x+^12 y=90 and so
l 3 ⊥l 4.
WhenδF(Z 1 ,Z 2 ,Z 4 )>0 we note that
D 1 ={Z∈Π:δF(Z 1 ,Z 2 ,Z)δF(Z 1 ,Z 4 ,Z)≤ 0 },
D 2 ={Z∈Π:δF(Z 1 ,Z 2 ,Z)δF(Z 1 ,Z 4 ,Z)≥ 0 },

and get a similar characterisation whenδF(Z 1 ,Z 2 ,Z 4 )<0.
Whenl 1 =l 2 ,wetakeD 1 =l 1 ; we could also takeD 2 =Πbut do not make any
use of this.