Geometry with Trigonometry

Sec. 3.5 Degree-measure of angles 43

Degree-measure has the further properties:-

(i)If∠BAC is a wedge-angle and D=AisinIR(|BAC),then|∠BAD|◦≤

|∠BAC|◦.If,further,D∈[A,Cthen|∠BAD|◦<|∠BAC|◦.

(ii)For non-collinear points A,B,CletH 1 be the closed half-plane with edge AB in

which C lies. If D=AisinH 1 and|∠BAD|◦≤|∠BAC|◦,thenD∈IR(|BAC).

A

B

C D H^1

Figure 3.9.

A

B

C D

E

H 1

H 4 H 3

Proof.
(i) AsD∈IR(|BAC),byA 5 (iii)|∠BAD|◦+|∠DAC|◦=|∠BAC|◦.ByA 5 (i),
|∠DAC|◦≥0so|∠BAD|◦≤|∠BAC|◦.

IfD∈[A,Cthen∠DACis not a null-angle, so|∠DAC|◦>0 and hence|∠BAD|◦<
|∠BAC|◦.
(ii) LetE=Abe such thatA∈[B,E].LetH 3 ,H 4 be the closed half-planes with
common edgeAC, withB∈H 3 andE∈H 4 .Then

H 1 =H 1 ∩Π=H 1 ∩(H 3 ∪H 4 )=(H 1 ∩H 3 )∪(H 1 ∩H 4 )

=IR(|BAC)∪IR(|EAC).

AsD∈H 1 , then eitherD∈IR(|BAC)orD∈IR(|EAC).
Now suppose thatD∈IR(|BAC),sothatD∈IR(|EAC)andD∈[A,C.By
A 5 (iii),

|∠EAD|◦+|∠DAC|◦=|∠EAC|◦.

Hence by A 5 (iii), as we have supplementary pairs of angles,

180 −|∠BAD|◦+|∠DAC|◦= 180 −|∠BAC|◦.

From this

|∠BAC|◦+|∠DAC|◦=|∠BAD|◦,

and as|∠DAC|◦>0, we have|∠BAC|◦<|∠BAD|◦. This gives a contradiction with
our hypothesis.