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

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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

[edit | edit source]Given two triangles and such that their sides are equal, hence:

- The side equals .
- The side equals .
- The angle equals (These are the angles between the sides).

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

## Proof

[edit | edit source]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.