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

The corresponding material in Euclid's elements can be found on page 33 of Book I, Proposition 4 in Issac Todhunter's 1872 translation, The Elements of Euclid for the Use of Schools and Colleges.

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:

The side ${\overline {AB}}$ equals ${\overline {DE}}$ .
The side ${\overline {CA}}$ equals ${\overline {DF}}$ .
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.

${\text{Superpose }}\triangle ABC{\text{ on }}\triangle DEF{\text{ such that }}A{\text{ is placed on }}D{\text{ and }}{\overline {AB}}{\text{ is placed on }}{\overline {DE}}$
$\because {\overline {AB}}={\overline {DE}}{\text{ (given)}}$
$\therefore B{\text{ coincides with }}E$
$\because \angle CAB=\angle FDE{\text{ (given)}}$
$\therefore {\overline {CA}}={\overline {FD}}$
$\because {\overline {CA}}={\overline {DF}}{\text{ (given)}}$
$\therefore C{\text{ coincides with }}F$
$\therefore {\overline {CB}}={\overline {EF}}$
$\therefore \triangle ABC=\triangle DEF$
$\therefore \triangle ABC\cong \triangle DEF$