34 Basic shapes of geometry Ch. 2
2.4 If[A,B],[C,D]are both segments of a linelsuch that[A,B]∩[C,D]=0, show/
that[A,B]∩[C,D]and[A,B]∪[C,D]are both segments.2.5 Show that ifA=BandC,Dare both inAB\[A,B], then either[A,B]∩[C,D]= 0 /
or[A,B]⊂[C,D].2.6 Let≤Ebe a total order on the setEandf:E→Fa 1:1 onto function. If for
a,b∈F,a≤Fbwhenf−^1 (a)≤Ef−^1 (b), show that≤Fis a total order onF.2.7 Use Ex.2.6 to show that ifFis an infinite set and there is a total order onF,
then there are infinitely many total orders onF.2.8 Show that interior regions have the following properties:-(i) IfP∈IR(|BAC)andP=A,thenAP∩IR(|BAC)=[A,P.
(ii) IfA,B,Care non-collinear andU∈[A,B,V∈[A,Cbut neitherUnorV
isA,thenUV∩IR(|BAC)=[U,V].
(iii) IfA,U,Vare distinct collinear points, andUandVare both in
IR(|BAC),thenV∈[A,U.2.9 Show that an exterior region has the following properties:-(i) The arms[A,Band[A,Care both subsets ofER(|BAC).
(ii) IfP∈ER(|BAC)andP=A,then[A,P⊂ER(|BAC).2.10 Show that convex quadrilaterals[A,B,C,D]have the following properties:-(i) Each of〈 2 〉[A,D,C,B], 〈 3 〉[C,B,A,D], 〈 4 〉[C,D,A,B],
〈 5 〉[B,A,D,C], 〈 6 〉[B,C,D,A], 〈 7 〉[D,A,B,C],
〈 8 〉[D,C,B,A],is equal to〈 1 〉[A,B,C,D].
(ii) Each of the verticesA,B,C,Dis an element of[A,B,C,D].
(iii) The linesACandBDcannot have distinct pointsEandFin common.
(iv) Each side and each diagonal is a subset of[A,B,C,D].
(v) Any pair of opposite sides are disjoint.2.11 For non-collinear pointsA,B,DletCbe a point which is in the interior region
IR(|BAD)but not in[A,B,[A,D or[A,B,D]. Show that thenA,B,C,Dare
the vertices of a convex quadrilateral.