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Example:Prove Triangles are Similar

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Similar triangles



Formally speaking, two triangles  and  are said to be similar if either of the following conditions holds:

1. Corresponding sides have lengths in the same ratio:

i.e. . This is equivalent to saying that one triangle is an enlargement of the other.

2.  is equal in measure to   and  is equal in measure to. This also implies that  is equal in measure to.

When two triangles  and  are similar, one writes

The 'is similar to' symbol can also be expressed as three vertical lines: lll

Angle/side similarities

The following three criteria are sufficient to prove that a pair of triangles is similar. In summary, they state that if triangles have the same shape then they are to scale (AA criterion), and that if they are to scale then they have the same shape (SSS). Another extra criterion, SAS, will also be explained below.

1.  AA: if two triangles have two corresponding pairs of angles with the same measure then they are similar. Sometimes this criterion is also referred to as AAA because two angles of equal measure imply equality of the third. This criterion means that if a triangle is copied to preserve the shape, then the copy is to scale. (Our solved example in mathguru.com uses this concept)


2.  SSS / SSS~ / Three sides proportional: If the ratio of corresponding sides of two triangles does not depend on the sides chosen, then the triangles are similar. This means that if any triangle copied to scale is also copied in shape.

3.  SAS / SAS~ / Ratio of two sides, included angle: if two sides are taken in a triangle, that are proportional to two corresponding sides in another triangle, and the angles included between these sides have the same measure, then the triangles are similar. This means that to enlarge a triangle, it is sufficient to copy one angle, and scale just the two sides that form the angle.



