Geometry with Trigonometry

46 Distance; degree-measure of an angle Ch. 3

We also get a contradiction iflcontains a pointP=AinIR(|B 1 AC)which is
not onAC. For then by 3.5.2|∠B 1 AP|◦<|∠B 1 AC|◦, so that 180−|∠BAP|◦< 180 −
|∠BAC|◦and so|∠BAC|◦<|∠BAP|◦. It follows from 3.5.2 that[A,C⊂IR(|BAP)
and so|∠BAC|◦+|∠CAP|◦=|∠BAP|◦.But|∠BAC|◦>0andso
|∠CAP|◦<|∠BAP|◦, which gives a contradiction.
Thuslmust contain a pointP=AinIR(|BAC).Asthen|∠BAP|◦+|∠PAC|◦=
|∠BAC|◦and|∠BAP|◦=|∠PAC|◦,wemusthave|∠BAP|◦=^12 |∠BAC|◦which deter-
mines[A,Puniquely.

Definition.Wedefinethemid-lineorbisectorof the angle-support |BACas
follows:- ifC∈[A,B then it is the lineAB, and otherwise it is the unique linel
just noted. We use the notation ml(|BAC)for this.

3.7 Degree-measure of reflex angles....................

3.7.1

Definition.Letαbe a reflex angle with support|BAC. We first suppose thatC∈AB,
and as in 3.5.2 let∠B 1 AC 1 be the vertically opposite angle of the wedge-angle∠BAC.
ThenB 1 ∈AC,C 1 ∈ABand∠B 1 ACis the vertically opposite angle for∠BAC 1 .By
3.5.2 we note that
180 +|∠B 1 AC|◦=|∠BAC 1 |◦+ 180 ,

and we define thedegree-measureofαto be the common value of these:

|α|◦= 180 +|∠B 1 AC|◦=|∠BAC 1 |◦+ 180.

Secondly, ifC∈[A,B so thatαis a full-angle, we define|α|◦=360.
Then for each reflex-angleα,|α|◦is defined; by 3.5.2 it satisfies 180<|α|◦< 360
unlessαis a full-angle in which case|α|◦=360.

 A

B

C 1

B 1

C

Figure 3.12. Measure of a reflex angle.

A

B

C 1

B 1

C

D

Letαbe a non-full reflex-angle with support|BACand take B 1 =A,C 1 =Aso
that A∈[B,B 1 ],A∈[C,C 1 ].Let[A,D ⊂IR(|B 1 AC 1 )but D∈[A,C 1 ,D∈[A,B 1.
Then
|∠BAD|◦+|∠DAC|◦=|α|◦.