3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Rotation, translation, reflection, contraction, and dilation. Poor shapes. It seems like any time they change, they get shunned.

Your students, on the other hand, won't be shunned for mastering the Angle-Angle criterion for establishing similarity between two triangles. To do that though, it only makes sense that they should be familiar with the notions of similarity, congruence, and different types of angles. (We're talking alternate interior angles, alternate exterior angles, corresponding angles, vertical angles, and more.) It will also behoove them to keep all of those theorems about congruent triangles in mind. You know, all the ones with S's and A's.

Students should also be familiar with the definitions—definishuns—of the similarity transformations that result in similar triangles: all the congruence transformations (translation, reflection, and rotation) and those that preserve shape (expansion and contraction).

The first three should be pretty familiar to students, who probably learned about them beginning back in kindergarten. They probably didn't know them by their fancy names, but slide, turn, and flip are pretty common vocab words for five year-olds, along with pasketti and petty larceny.

If students are having a hard time remembering which transformation is which, they might secretly enjoying getting out of their seats to act out each change. Make it a game and have their peers guess which transformation they're performing. You might just be amazed at how talented their performances can be.

Demonstrating mastery of the Angle-Angle similarity postulate requires a little talent as well. Students will have to be able to identify two pairs of congruent angles in two triangles. They should also know that since all the angles in a triangle add up to 180°, knowing two angles actually means knowing three. Sometimes it's as easy as calculating the measure of the unlabeled angle, but more often than not, students will need to draw on their knowledge of theorems about congruent, complementary, and supplementary angles.

Make sure that they're up to speed on all those theorems, and that they know the rules for writing proofs, and they should be good to go.