Geometry for Elementary School/The Side-Angle-Side congruence theorem

In this chapter, we will discuss another congruence theorem, this time the Side-Angle-Side theorem. The angle is called the included angle.

The Side-Angle-Side congruence theorem
Given two triangles $$\triangle ABC $$ and $$\triangle DEF $$ such that their sides are equal, hence:
 * 1) The side $$\overline {AB} $$ equals $$\overline {DE} $$.
 * 2) The side $$\overline {CA} $$ equals $$\overline {DF} $$.
 * 3) The angle $$\angle CAB $$ equals $$\angle FDE $$ (These are the angles between the sides).

Then the triangles are congruent and their other angles and sides are equal too. Success!

Proof
We will use the method of superposition – we will move one triangle to the other one and we will show that they coincide. We won’t use the construction we learnt to copy a line or a segment but we will move the triangle as whole.


 * 1) $$\text{Superpose } \triangle ABC \text{ on } \triangle DEF \text{ such that } A \text { is placed on } D \text { and } \overline {AB} \text{ is placed on } \overline {DE}$$
 * 2) $$\because \overline {AB} = \overline {DE} \text{ (given)}$$
 * 3) $$\therefore B \text{ coincides with } E $$
 * 4) $$\because \angle CAB = \angle FDE \text{ (given)}$$
 * 5) $$\therefore \overline {CA} = \overline {FD} $$
 * 6) $$\because \overline {CA} = \overline {DF} \text{ (given)}$$
 * 7) $$\therefore C \text{ coincides with } F $$
 * 8) $$\therefore \overline {CB} = \overline {EF} $$
 * 9) $$\therefore \triangle ABC = \triangle DEF $$
 * 10) $$\therefore \triangle ABC \cong \triangle DEF $$

Geometria per scuola elementare/Il teorema di congruenza lato-angolo-lato