Investigating AAS
Objective: At the end of this lesson, students will understand the idea of a “counterexample”.Students will conduct an investigation to show AAS is sufficient to prove triangles are congruent, and then understand the proof of this conjecture.
Instructional Materials: Ruler/straight edge (index card works well), black marker, patty paper (squares of tissue paper)
1.Review SSS and SAS.Show other possibilities such as AAA and AAS and discuss how some of these are not enough to show two triangles are congruent.
2.Discuss the concept of counterexample.(Give non-math examples such as “All birds can fly”.We know that this is not true by giving only one counterexample—penguin. ) We do not have to go through every case—one counterexample is enough to prove that something is NOT true.

3.Show counterexample for AAA and SSA.A

2 equilateral triangles contradict AAA
If c is the center of a circle, let side BC be fixed, side AC is fixed and angle B is fixed (SSA). However, if the circle is drawn with AC as the radius, we see that the same radius will meet with segment AB (AB crosses through the circle and thus, intersects it twice) whose length was not fixed in the SSA conjecture.So now we have two different triangles with the same two sides and angle.
4.Explain unincluded side by comparing ASA and AAS.
When you have two angles and a side between them as in the case of ASA, this is an “included side”.For AAS we need to use any side that is not between the two angles, an “unincluded side”.

5.Take a vote to see who thinks AAS is enough to prove that two triangles are congruent.

6.Explain how we are going to investigate in pairs AAS and pass out supplies and worksheet.They will try to construct a counterexample.

7.Walk around, monitor class, answer questions.

Tip: Students may have trouble understanding that the rays that make an angle can be extended—that those side lengths are not fixed. Remind them of the plain, non-colored fettuccine from the previous investigation.

8.Discuss how the investigation points to AAS being enough, but it does not prove it. So…

9.Set up a proof of AAS—it uses ASA!

Given angle-angle-side, we will show that  two triangles are congruent. Using the angle sum theorem, we know that the third angle of both triangles must    also be congruent.Now we have two angles and the included side of two triangles congruent.Well, this is exactly the right set up for ASA! So these triangles are congruent by ASA, but we were only given AAS.Thus, only given angle- angle -side is sufficient to prove that two triangles are congruent.
10.Assign homework and journal topic. 


Connections to Standards for High School Mathematics

This lesson meets the standards for Geometry as students “investigate and validate conjectures”, specifically if given two angles and an unincluded side is sufficient to prove that two triangles are congruent.This lesson also provides an opportunity for students explore relationships between geometric figures, specifically the relationship of congruence.The activity strengthens the students’ skills to investigate conjectures and use various types of reasoning and proof.These skills are named under the process standard of Reasoning and Proof in which students learn to develop math arguments and proofs and also in the ISBE Standards.This lesson also uses assessment that emphasizes communication skill in mathematics.Students are required to keep a journal to explain their mathematical reasoning and record their investigations.In addition open-ended questions are asked to guide students to use the language of math to communicate their thinking.This process is consistent with both the NCTM Standards and the Illinois State Standards for communication.
    Rationale for Assessment
The assessment measures used in this lesson check for deeper understanding of concepts and require students to communicate mathematical reasoning.By assigning a journal, the teacher can see what thought processes the student is actually going through as they learn new concepts.Often, this is more beneficial for a teacher to see how well her students are doing, than checking for the right process in an objective type item.On the homework assignment, there are open-ended questions where again the student communicates mathematically as he or she argues whether triangles are congruent or not.In doing this they are also developing a sense for arguing and reasoning mathematically, consistent with the NCTM Principles and Standards for High School Mathematics.

Accommodations for Diverse Learners

This lesson has many features that would cater to the needs of diverse learners.For example, the lesson itself consists of very visual information including pictures and diagrams that would help meet the needs of visual learners.For the kinesthetic learners, the activity is very hands-on and the students can manipulate triangles in their investigation.To accommodate for those students that may struggle to catch everything from merely listening, all students receive a handout of instructions that give a step-by-step process to follow through the investigation.Pairing the students together is also a great way to help diverse learners that may need assistance or collaboration when working on such an open-ended question.

Day 4 - Congruent Triangles 1              Teacher Component Page            Day 6 - Similar Triangles