CONCEPT 1 – Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
When studying congruent triangles in G.CO.8 we established some minimum requirements that would guarantee congruence through a single or sequence of isometric transformations. We found that SSS, SAS, ASA, AAS and HL (and some special cases of ASS) to be enough information to always establish congruence between two triangles. In a likewise manner, we want to find the minimum requirements in two triangles to establish similarity. The way we are going to do this is to use a single or sequence of similarity transformations that would map one triangle onto the other.
|To establish the basic criterion for similarity we perform similarity transformations. You will see here how we perform a dilation to make the side congruent and then perform an isometric transformation to map them onto each other. These aren't that difficult and they reinforce earlier concepts.
CONCEPT 2 – Use the properties of similarity transformations to establish the SAS criterion for two triangles to be similar.
Now let us see if knowing two corresponding proportional sides and the included corresponding congruent angle (SAS) is enough for establishing similarity. Just as it was important in congruence the angle must be the included angle – meaning that it is the one between the two sides.
CONCEPT 3 – Use the properties of similarity transformations to establish the SSS criterion for two triangles to be similar.
|The proofs coming are very traditional in their format. There is nothing wrong with that - once we have established the criterion through transformational approaches we can then use them in a two column proof or paragraph proof.
CONCEPT 4 – Use the similarity criteria of AA, SAS and SSS to prove triangles to be similar.
Sometimes we prove similarity to establish new relationships about the triangle. Once similarity is established we know that there are three corresponding congruent angles and that the sides are proportional.
This is a bit like congruence (G.CO.8) where once we had proven the triangles to be congruent we could use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to gather new relationships about the triangle.