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

 Geometry for Elementary School The Side-Side-Side congruence theorem The Side-Angle-Side congruence theorem The Angle-Side-Angle 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. $\therefore \triangle ABC \cong \triangle DEF$