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
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.