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Example:Prove Triangles are Similar

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Two geometrical objects are called similar if they both have the same shape. More precisely, one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly "stretching" the same amount on all directions, possibly with additional rotation and reflection, i.e., both have the same shape, and one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar.


Similar triangles

To understand the concept of similarity of triangles, one must think of two different concepts. On the one hand there is the concept of shape and on the other hand there is the concept of scale.

In particular, similar triangles are triangles that have the same shape and are identical to one another except for scale. For a triangle, the shape is determined by its angles, so the statement that two triangles have the same shape simply means that there is a correspondence between angles that preserve their measures.

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.



