Geometry for Elementary School/Copying a triangle

< Geometry for Elementary School

In this chapter, we will show how to copy a triangle $\triangle ABC$ to other triangle $\triangle CDE$ . The construction is an excellent example of the reduction technique – solving a problem by solution to a previously solved problem.

The construction

Construct a triangle from the sides of $\triangle ABC$ : $\overline {AB}$ , $\overline {AC}$ , $\overline {BC}$ and get $\triangle CDE$ .

Claim

The triangles $\triangle ABC$ and $\triangle CDE$ are congruent.

Proof

$\overline {AB}$ , $\overline {AC}$ , $\overline {BC}$ are sides of the triangle $\triangle ABC$ and therefore obey the triangle inequality.
Therefore one can build a triangle whose sides equal these segments.
The sides of the triangle $\triangle ABC$ and $\triangle CDE$ are equal.
Due to The Side-Side-Side congruence theorem the triangles $\triangle ABC$ and $\triangle CDE$ congruence.

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