Sec. 4.3 Properties of triangles and half-planes 59
4.2 There is an AAS-principle of congruence that if|∠BAC|◦=|∠EDF|◦,|∠CBA|◦=|∠FED|◦,
|B,C|=|E,F|,then the triangles [A,B,C],[D,E,F] are congruent. [Hint. Suppose that
|∠BCA|◦<|∠EFD|◦; lay off an angle∠BCGequal in magnitude to∠EFD
and withGon the same side ofBCasAis; then[C,Gmeets[A,B]at a pointH;
also[H,B,C]≡[D,E,F]and in particular|∠BHC|◦=|∠EDF|◦=|∠BAC|◦;
deduce a contradiction and then apply the ASA-principle.]4.3 There is an ASS-principle of congruence for right-angled triangles, that ifBC⊥
BA,EF⊥ED,|C,A|=|F,D|,|A,B|=|D,E|,then[A,B,C]≡[D,E,F].[Hint.
TakeC′so thatE∈[F,C′]and|E,C′|=|B,C|.]4.4 IfP∈ml(|BAC)andQ=πAB(P),R=πAC(P),then|P,Q|=|P,R|. Conversely,
ifP∈IR(|BAC)and|P,Q|=|P,R|whereQ=πAB(P),R=πAC(P),then
P∈ml(|BAC).4.5 In triangles[A,B,C],[D,E,F]let|A,B|=|D,E|,|A,C|=|D,F|,|∠BAC|◦>|∠EDF|◦.Then|B,C|>|E,F|. [Hint. Lay off the angle∠BAGwith|∠BAG|◦=|∠EDF|◦
and withGon the same side ofABasCis. IfG∈BCproceed; ifG∈BC,let
K=mp(G,C)and show that[A,Kmeets[B,C]in a pointH.]4.6 IfAB‖AC,thenAB=AC.4.7 LetH 1 be a closed half-plane with edgel,letP∈H 1 and letO=πl(P).Then
ifmis any line inΛsuch thatO∈m,wemusthaveπm(P)∈H 1.4.8 With the notation of 4.1.1 show that in all casesT(A,B,C)→≡(A,B,C)Tand if
T(A,B,C)→≡(A′,B′,C′)T′thenT′(A′,B′,C′)→≡(A,B,C)T. Show also that if
T(A,B,C)→≡(A′,B′,C′)T′andT′(A′,B′,C′)→≡(A′′,B′′,C′′)T′′,thenT(A,B,C)→≡(A′′,B′′,C′′)T′′.
Thus congruence of triangles is reflexive, symmetric and transitive, and thus is
an equivalence relation.