Explanation:
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.
http://en.wikipedia.org/wiki/Similarity_(geometry)
The above explanation is copied from
Wikipedia, the free encyclopedia and is remixed as allowed under the Creative
Commons Attribution- ShareAlike 3.0 Unported License.