These three theorems, known as Angle - Angle (AA), Side - Angle - Side (SAS), and Side - Side - Side (SSS), are foolproof methods for determining similarity in triangles. Similar triangles are easy to identify because you can apply three theorems specific to triangles. Next, compare the ratios of the lengths of the sides that include A and F. Note: Note that in similar triangles, each pair of corresponding sides are proportional.Īlso, if two triangles are congruent, therefore they are similar (although the converse is not always true). 2 EXAMPLE 3 Use the SAS Similarity Theorem Both m A and m F equal 53, so A F. Example 2Let the vertices of triangles ABC and PQR defined by the coordinates: A(-2,0), B(0,4), C(2,0), P(-1,1), Q(0,3), and R(1,1). If two sides of a triangle are proportional to two sides of a second triangle, and their included angles are. $\Rightarrow$\, since we know that if two triangles are congruent, therefore they are similar. The SAS Similarity Theorem states that two triangles are similar if the lengths of two sides of one triangle are proportional to the lengths of two. Side-Side-Side (SSS) Similarity Theorem If the three sides of a triangle are proportional to the corresponding sides of a second triangle, then the triangles are similar. There are three ways to find if two triangles are similar: AA, SAS and SSS: AA. Thus, if A X and AB/XY AC/XZ then ABC XYZ. Therefore, by the SAS Congruency Criterion, two or three out of the six is usually enough. SAS or Side-Angle-Side Similarity If the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then two triangles are said to be similar.
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