Geometry with Trigonometry

Sec. 3.7 Degree-measure of reflex angles 47

Proof.As[A,D ⊂IR(|B 1 AC 1 ),Dis in the closed half-plane with edgeAB
in whichC 1 lies, and also in the closed half-plane with edgeACin whichB 1 lies.
By 3.5.2,|∠B 1 AD|◦<|∠B 1 AC 1 |◦so by A 5 (iii)|∠BAC 1 |◦<|∠BAD|◦. By 3.5.2 then
[A,C 1 ⊂IR(|BAD), and by similar reasoning[A,B 1 ⊂IR(|DAC).Then

|∠BAD|◦+|∠DAC|◦=|∠BAD|◦+(|∠DAB 1 |◦+|∠B 1 AC|◦)

=(|∠BAD|◦+|∠DAB 1 |◦)+|∠B 1 AC|◦

= 180 +|∠B 1 AC|◦=|α|◦.

COMMENT. We could use this last result to employ the measures of reflex-angles
to a significant extent, but in fact do not do so until our full treatment of them in
Chapter 9.

Exercises

3.1 IfB∈[A,C],then|A,B|≤|A,P|≤|A,C|for allP∈[B,C].

3.2 LetA,B,Cbe points of a linel,andM=mp(A,B).IfCisAorB,orifC∈

l\[A,B],then|C,A|+|C,B|= 2 |C,M|.

3.3 LetA,B,Cbe distinct points andD=mp(B,C),E=mp(C,A),F=mp(A,B).

Prove thatD,E,Fare distinct. IfA∈BC, show that neitherEnorFbelongs to

BC.

3.4 IfA=B, show that

{P∈AB:|B,A|+|A,P|=|B,P|}

is the half-line ofABwith initial-pointAwhich does not containB, while

[A,B={P∈AB:|A,P|+|P,B|=|A,B|or|A,B|+|B,P|=|A,P|}.

3.5 Find analogues of 3.3.1 whenA∈[B,C]and whenB∈[C,A].

3.6 Show that ifA,B,C,Dare distinct collinear points such thatC∈[A,B],

B∈[A,D],and

|A,C|

|C,B|

=

|A,D|

|D,B|

,

then

1

|A,C|

+

1

|A,D|

=

2

|A,B|

.

3.7 Show that ifA,B,Care non-collinear points, andP=Ais a point of

IR(|BAC),then

IR(|BAP)∪IR(|PAC)=IR(|BAC),IR(|BAP)∩IR(|PAC)=[A,P.