**Theorem Pythagoras’ Theorem**

In a right – angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.

**Theorem (Sine Rule)**

In a triangle where ; ; are the lengths of the sides opposite the vertices ; ; , respectively,

**Theorem (Cosine Rule)**

In a triangle where ; ; are the lengths of the sides opposite the vertices , , respectively,

**Congruence of triangles**

Triangles may be determined to be congruent by any of the following rules.

Note that when we say two triangles and are congruent we mean that the correspondence of vertex to , to and to determines the congruence. We denote that two triangles and are congruent by writing .

**Similarity of triangles**

Each of the congruence rules has a corresponding similarity rule, by replacing side-length equality by proportionality. Thus, triangles may be determined to be similar by any of the following rules.

As with congruence, when we say two triangles and are similar we mean that the correspondence of vertex to , to and to determines the similarity. We denote that two triangles and are similar by writing .

**Theorem**

If a line joins the midpoints of two sides of a triangle then that line is parallel to the third side and its length is equal to one half of the length of the third side.

**Theorem**

A line parallel to one side of a triangle divides the other two sides in the same proportion.

**Theorem**

The angle bisector of one side of a triangle divides the opposite side in the same ratio as the other two sides.