Geometry for Elementary School/Copying a triangle

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

1. 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

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