Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
INTERPRETATION OF OBJECTIVE - G.SRT.A.3
Again paralleling the same process we did with congruence, first we define the motion in the plane, the dilation (G.SRT.A.1), then we look at what it means to be similar (G.SRT.A.2) and then finally we will determine the similarity criteria. This objective only mentions the AA criteria but I feel that we should look at SAS and SSS proportionality criteria as well.
SKILLS
(1) The student will be able to prove two triangles to be similar using the minimum requirements of AA, SAS and SSS.
(2) The student will be able to use the properties of similarity transformations to establish the AA, SAS and SSS criterion for two triangles to be similar.
THE BIG IDEA
Other than congruence... there is no bigger idea than similarity. Ratios in diagrams constantly show up and at the heart of those most often is the hidden similarity relationship. Similarity shows up in all areas from this point forward - partitioning a segment, angle values in a circle, parallel lines, mid-segments, etc... the list just keep going on and on. Recognizing and using the concepts of similarity is essential for success in geometry.
TRAPS & PITFALLS
When we did proof of congruence we looked for 'hidden' items in the diagram that weren't specifically mentioned in the given such as vertical angles, a common side, angles formed by parallel lines, etc... These same items help us succeed in proof of similarity. The one that is most 'hidden' to the student eye is the common angle. A very common setup for similar triangles is to have an overlapping angle. So in the diagram below ∠A is common in both triangles - this is often missed by students.
PAST CONNECTIONS
In G.CO.A.8 we looked at the criteria for triangle congruence, and now we are doing the same thing but with establishing similarity between figures. Much of the same logic and work takes place for both of these processes.
FUTURE CONNECTIONS
As mentioned in the big idea, establishing figures to be similar is a very common theme throughout the rest of the year. Similarity of triangles shows up very very often because the only required knowledge is two corresponding congruent angles, and this happens in many different cases, it happens almost automatically anytime parallel lines are present in the diagram.
MY REFLECTIONS (over line l)
Nothing really hidden in this objective, we have been teaching similar proof and specifically AA from the beginning of time. I found the students to picked it up quickly because how 'similar' it is to what we did with congruence.
Most proofs of this type are AA and not SAS or SSS even through they are viable ways to also establish similarity. I think they are not used as much because to get the information into the diagram it is difficult, it would probably require giving lengths and I can't recollect ever seeing that. Usually the given expressly gives that the ratios exist which takes any adventure out of it.
