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Similarity of Triangles
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In general, we come across several objects which have something common between them.  Observing them closely, we can see that some of them have same shape but may have different or same size.

For example, if we consider the photographs a person developed from same negative, they all look same in all respect except for their size.  Such objects are called similar objects

Two line segments of different sizes, two circles of different radii, two squares of different sizes, two rectangles of different dimensions – come under similar figures.  One smaller circle can be got by shrinking a larger circle.  One bigger square can be got by stretching a smaller square. Then, what about the similarity of triangles? Is it true to say any two given triangles are similar?  The answer is NO.   This is true only when the  triangles are equilateral. For all other triangles, we have the following statement which lays down the condition for the similarity of two triangles.

“Two triangles are said to be similar if their corresponding angles are equal and their sides are proportional

We use the symbol  for the similarity of two triangles.  We write for the similarity of and .  Further, we follow that the vertices, the angles and
. This ratio is called Scale factor.

Basic proportionality theorem or Thales’ theorem: If a line is drawn parallel to one side of a triangle and it intersects the other two sides in two distinct points then it divides the two sides in the same ratio.

In the , if DE||BC, then.

Converse of Basic Proportionality Theorem: If a line divides any two sides of a triangle in the same ratio, the line must be parallel to the third side.
In the, if D and E are two points on AB and AC respectively such that,, then DE||BC