I am in the unit on Triangle Congruence and I am having a hard time helping students "see" the included side with ASA and distinguish between ASA and AAS. They have not struggled with the included angle for SAS, but the included side and a side not included is really tripping them up. Does anyone have a solution?
Let the students come up with their own problem by drawing a triangle each and suggest angles and sides and then try to find the missing angle or side in their cases. This way they will realise that it is immposible in some cases to find answers except in certain rules which include the SAS and the AAS.
Have you tried giving examples on each case? Give them example to prove all cases both right and wrong. Or you could try and challenge them of coming up with the congruent triangle for ASA and AAS triangles.
I use hands-on activities using straws and pipe cleaners to learn the triangle congruence theorems. For ASA, I have students (in pairs) cut pieces of two straws to specific lengths and create a triangle using the two fixed lengths and a fixed included angle (which you will specify and they will measure out with a protractor). When they are satisfied that they have met all the requirements, they will attach a third side (straw) to complete the triangle. Then, students will compare triangles to see that they have created congruent triangles using ASA. You can do a similar activity for AAS, but you have to bend the pipe cleaner several times to get the 'third point' to meet to form a triangle. This is also a powerful way to make students see how SSA does not work because it is possible to create two different triangles using the straws and pipe cleaners.
The included side is an ASA postulate leads to congruency of the two triangles. Like that SAS and SSS are postulates that determine triangles uniquely. Hence these are postulates for congruency.
The not included sides and an angle or SSA or AAS are not uniquely determining the triangles . So there could be two different non congruent triangles in this context. Hence there are no postulates like SSA or AAS for congruency of triangles. These need to be taught in mathematical projects or mathematical practical classes through Matlabs, till the young mind gets acquainted with . The grasping ,visualising and process of stabilisation of such ideas need not be quick with many students.We need repeated patient visual confirmations for the idea to percolate deep in mind to settle and stabilise.
Sometimes color coding them works well - drawing the angles and side in the same color and they can see that the three are connected for ASA, but more spread out for AAS.
I have also found that if I tell them to put their pencil on one angle and and trace over the side to the other angle that they know, it's ASA if the side they traced is the side that's congruent on the other triangle. If they traced a side that they didn't know was congruent to one in the other triangle then it's AAS.