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Geometry: Congruence

Similarity

Problems

Problems

Similarity is much like congruence, except in order for polygons to be similar, they only need to have the same shape. As we did with congruence, we will study similarity in triangles to simplify things. Formally speaking, two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. For example, if triangles ABC and DEF are similar, then angle pairs AB and DE, BC and EF, and CA and FE are all equal. Also, AB/DE=BC/EF=CA/FD. If these three ratios are equal, then the corresponding sides are said to be proportional.

Figure %: Triangles ABC, DEF, and HIJ are similar

An easy to way to create similar triangles is by drawing a line through a triangle such that it intersects with two sides and is parallel to the third side. This new line will form a new triangle that is smaller that the original, but similar to it.

Figure %: Similar triangles
The line l , parallel to AC, creates the triangle DEB, which is similar to triangle ACB. Another thing about the above diagram: because the two triangles ACB and DEB are similar, DB/AB = EB/CB. We also know that AD/DB = CE/EB. This last relationship is just another application of similarity in triangles. In the next lesson we'll see how to prove triangles are simi lar.

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