Geometry with Trigonometry
40 Distance; degree-measure of an angle Ch. 3 In turn s= r 1 −r and so s−sr=r, giving s=r( 1 +s) and thus r= s 1 +s 3.4 Thecross ...
Sec. 3.5 Degree-measure of angles 41 AXIOM A 5 .Degree-measure||◦of angles has the following properties:- (i)In all cases|α|◦≥0; ...
42 Distance; degree-measure of an angle Ch. 3 A B C H 1 k Figure 3.7. Laying off an angle. A B C B 1 C 1 Figure 3.8. Oppos ...
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) ...
44 Distance; degree-measure of an angle Ch. 3 3.6 Mid-line of an angle-support 3.6.1 Right-angles Definition. Given any pointP= ...
Sec. 3.6 Mid-line of an angle-support 45 3.6.3 Mid-lines................................ Given any angle-support|BACsuch that C ...
46 Distance; degree-measure of an angle Ch. 3 We also get a contradiction iflcontains a pointP=AinIR(|B 1 AC)which is not onAC. ...
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 li ...
48 Distance; degree-measure of an angle Ch. 3 3.8 Show that ifdis any positive real number and||is a distance function, then d|| ...
4 Congruence of triangles; parallel lines COMMENT. In this chapter we deal with the notion of congruence of triangles, and make ...
50 Congruence of triangles; parallel lines Ch. 4 We say thatTiscongruenttoT′, writtenT≡T′,ifTis congruent toT′in at least one of ...
Sec. 4.1 Principles of congruence 51 (ii) Note that if T 1 ,T 2 are the triangles with vertices {A,B,D},{A,C,D}, respec- tively, ...
52 Congruence of triangles; parallel lines Ch. 4 Then by the SAS principle,T(B,C,A)→≡(B′,C′,D′)T′′. In particular |∠C′B′D′|◦=|∠C ...
Sec. 4.2 Alternate angles, parallel lines 53 CASE 2. LetB∈[E,C].Then[A,B ⊂IR(|DAC)and[D,B∈IR(|ADC).It follows that |∠BAC|◦=|∠DAC ...
54 Congruence of triangles; parallel lines Ch. 4 COMMENT.IfD and H are on opposite sides of BC,then∠CBH and ∠BCDare known asal- ...
Sec. 4.3 Properties of triangles and half-planes 55 Proof. (i) and (ii) follow immediately from the definition, while (iii) foll ...
56 Congruence of triangles; parallel lines Ch. 4 by 4.1.1, so by our last result,|A,D|>|A,C|.However|A,D|=|A,B|+|B,D|as B∈[A, ...
Sec. 4.3 Properties of triangles and half-planes 57 Proof. Existence.LetA,Bbe distinct points ofl. Take a pointQon the opposite ...
58 Congruence of triangles; parallel lines Ch. 4 l πl(P) P Figure 4.13. Projection to the linel. l sl(P) P Axial symmetry i ...
Sec. 4.3 Properties of triangles and half-planes 59 4.2 There is an AAS-principle of congruence that if |∠BAC|◦=|∠EDF|◦,|∠CBA|◦= ...
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