4 Congruence of triangles; parallel lines
COMMENT. In this chapter we deal with the notion of congruence of triangles, and
make a start on the concept of parallelism of lines. As we have distance and angle-
measure, we do not need special concepts of congruence of segments and congruence
of angles, and we are able to definecongruence of triangles instead of having it as a
primitive term as in the traditional treatment. As a consequence there is a great gain
in effectiveness and brevity.
4.1 Principles of congruence
4.1.1 Congruence of triangles
A
x
B
y C
z
A′
x
B′ y
C′
z
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Figure 4.1. Congruent triangles.
Definition.LetTbe a triangle with the vertices{A,B,C}andT′a triangle with
vertices{A′,B′,C′}. We say thatTiscongruenttoT′ in the correspondenceA→
A′,B→B′,C→C′,if
|B,C|=|B′,C′|, |C,A|=|C′,A′|, |A,B|=|A′,B′|,
|∠BAC|◦=|∠B′A′C′|◦,|∠CBA|◦=|∠C′B′A′|◦,|∠ACB|◦=|∠A′C′B′|◦.
We denote this byT(A,B,C)→≡(A′,B′,C′)T′.
Geometry with Trigonometry
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