Geometry with Trigonometry

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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http://dx.doi.org/10.1016/B978-0-12-805066-8.50004-5