1) definition of isometry 2) isometries as congruence mappings (triangles are mapped to congruent triangles, etc) 3) an isometry is determined by the image of any three non-collinar points 4) rotations, translations, reflections, glide reflections; definitions and properties of these types 5) theorem: any isometry is a composition of at most three reflections 6) representing given isometry as a composition of reflections as above (how many refections do you need to represent rotations, translations, glide reflections? Can you represent in this way an isometry which is, say, a composition of given two rotations? or a rotation and translation? 7) classification theorem: any isometry belongs to one of the four types (rotations, translations, reflections, glide reflections) Determining the type of a given isometry (for example, what is a composition of a rotation and a glide reflection?) ---------------------------------------------------------------------------- Note that there are a few very helpful tricks you can use in the questions above. The first one is #3 in the list above: if, for example, you can see that your isometry transforms some three non-collinear the way a particular rotation does, then isometry *must* be equal to that rotation, and there's no need to check all the other points. Another trick is to paint two "sides" of the plane in different colors, and to see whether the isometry flips the colors or not (reflections and glide reflections do, rotations and translations don't). This way you can see, for example, that a composition of six reflections can only be a rotation or translation, but never a reflection. The last trick is to decompose all isometries into reflections, choosing the axes of reflections in a clever way, so that some of the reflections cancel out when you compose your isometries. We did this, for example, when studying compositions of two rotations. More on all these tricks is in the lecture notes. ----------------------------------------------------------------------------- 8) Commensurable and incommensurable segments, common measure, the Euclidean algorithm, ratio of two segments 10) Thales' theorem (aka lemma in section 159 in the book & its corollaries) 9) Definition of similarity mappings. Homothety is a similarity mapping. 10) Important theorem: any similarity mapping is a composition of an isometry and a homothety; the center of homothety can be chosen where you wish. 11) Properties of similarity mappings: straight lines are mapped to straight lines, circles to circles, etc. 12) Definition of similar figures. Proving that any two segments are similar, any two circles are similar, etc. (This means finding a specific similarity mapping that maps one segment to the other, etc.) 13) Similar triangles and tests for similarity. (A very useful trick for #11-13 is using theorem from #10 (this is described in detail in lecture notes.) 14) Construction problems (using isometries -reflections, rotations, translations - and similarity/homotheties.) This is very important, please take a careful look at the examples in the book. You can do more questions from the book for extra practice.