Sec. 2.3 Angle-supports, interior and exterior regions, angles 29
Definition. We call a pair{[A,B,[A,C}of co-initial half-lines anangle-support.
For this we use the notation|BAC.WhenA∈[B,C], this is called astraight angle-
support. We call the half-lines[A,Band[A,Cthearms, and the pointAthevertex,
of|BAC. Note that we are assumingB=AandC=Afrom the definition of half-lines.
In all cases we have|BAC=|CAB.
COMMENT. The reason that we do not call|BACanangleis that there are two
angles associated with this configuration.
A
B
C
Figure 2.7. An interior region.A
B
C
The corresponding exterior region.Definition. Consider an angle-support|BACwhich is not straight. WhenA,B,C
are not collinear, letH 1 be the closed half-plane with edgeABin whichClies, and
H 3 the closed half-plane with edgeACin whichBlies. ThenH 1 ∩H 3 is calledthe
interior region of|BAC, and we denote it byIR(|BAC).WhenA,B,Care collinear
we have[A,B=[A,Cand we defineIR(|BAC)=[A,B.
Interior regions have the following properties:-(i)[A,B and[A,C are both subsets ofIR(|BAC).(ii) If P,Q∈IR(|BAC)then[P,Q]⊂IR(|BAC), so that an interior region is a
convex set.(iii)If P∈IR(|BAC)and P=A, then[A,P⊂IR(|BAC).Proof.
(i) WhenA,B,Care non-collinear, by 2.1.5[A,B⊂AB⊂H 1 andby2.2.3[A,B⊂
H 3 so[A,B⊂H 1 ∩H 3. Similarly for[A,C.When[A,B=[A,Cthe result is trivial.
(ii) WhenA,B,Care non-collinear, we have that[P,Q]is a subset ofH 1 by 2.2.3.
It is a subset ofH 3 similarly, and so is a subset of the intersection of these closed
half-planes. When[A,B=[A,C, the result follows from 2.1.5.
(iii) WhenA,B,Care non-collinear, by 2.2.3[A,Pis a subset of each ofH 1 and
H 3 , and so of their intersection. When[A,B=[A,Cwe haveIR(|BAC)=[A,B
and[A,P=[A,B.