Similarity Criteria

Criteria for Similar Triangles 

1. AAA Similarity - If two triangles are equiangular, then they are similar. i.e. , If in two triangles the corresponding angles are equal then, their corresponding sides are proportional and hence the triangles are similar. 

2. SSS Similarity - If the corresponding sides of two triangles are proportional then their corresponding angles are equal and hence the triangles are similar. 

3. SAS Similarity - If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional then triangles are similar. 

4. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then the triangles on both sides of the perpendicular are similar to the whole triangle and also to the each other. 

5. If two triangles are equiangular, then ratio of their corresponding sides is the same as the ratio of the corresponding altitudes. 

6. If two triangles are equiangular, then the ratio of their corresponding sides is the same as the ratio of the corresponding medians. 

7. If two triangles are equiangular, then the ratio of the corresponding sides is the same as the ratio of the corresponding angle - bisector segments. 

8. If one angle of a triangle is equal to one angle of another triangle and the bisectors of these equal angles divide the opposite sides in the same ratio then triangles are similar. 

9. If two sides and a median bisecting one of these sides of a triangle are respectively proportional to the two sides and the corresponding median of another triangle, then the triangles are similar. 

10. If two sides and a median bisecting the third side of a triangle are respectively proportional to the corresponding sides and the median of another triangle, then two triangles are similar.