CONCEPT 1 - Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motion.
One way to establish the criteria for triangle congruence to is to construct triangles based on given information and see if they will always be congruent to each other. This is not the best technique because how would you know if you created all possible triangles under those circumstances and unless you tried all cases you wouldn’t know for sure. That approach is great for giving us an idea of what we can expect or in some case rules certain things out but a more formal proof is need to establish it for all situations.
|TEACHER NOTE -- A great way to establish these relationships is by discovery. One of the activities in this objective takes students through the discovery process. It doesn't represent proof, but it does solidify the concepts in a meaningful way.
DOES SSS ESTABLISH TRIANGLE CONGRUENCE? YES!!!
DOES SAS ESTABLISH TRIANGLE CONGRUENCE? YES!!!
DOES ASA ESTABLISH TRIANGLE CONGRUENCE? YES!!!
DOES AAS/SAA ESTABLISH TRIANGLE CONGRUENCE? YES!!!
Yes, this is a special form of ASA. In any triangle, if you have two angles you are able to determine the third angle because they sum to 180°. So by knowing two angles, you actually know three angles. So ASA and AAS/SAA both work as congruence relationships.
|TEACHER NOTE -- I include the ASS/SSA congruence criteria cases because if you are teaching honors you need to later teach the Sine Law and its ambigious cases. Knowing about ASS/SSA will help them later gain a fairly complex idea. If you are not doing honors - this is not essential.
DOES ASS/SSA ESTABLISH TRIANGLE CONGRUENCE? SOMETIMES…..
This is a difficult case…. it sometimes produces congruence and other times it doesn’t. We will look at each case to see when it works and when it doesn’t.