FindXYandm^6 X.
Answer:XY=12 cm andm^6 X=43 degrees.
Triangle Congruence Using ASA and AAS
An Important Distinction –At first students may not see why it is important to identify whether ASA or AAS is the
correct tool to use for a specific set of triangles. They both lead to congruent triangles, right? Yes, but this will not
always be the case, as they will see in the next lesson. Sometimes the configuration of the corresponding congruent
sides and angles in the triangles determines if the triangles can be proved to be congruent or not. Knowing this will
motivate students to study the difference between ASA and AAS.
Flowchart Proofs –Flowchart proofs do a much better job of showing implication than two-column proofs. In a
two-column proof one statement following another does not necessarily mean that the previous statement implies
the next. Sometimes all the given information is listed at the beginning or another parallel argument needs to be
developed before the implication is made. This can be confusing for students without much experience with proofs,
or who have trouble understanding the argument. In a flowchart proof the implications are clearly indicated with
arrows, and when parallel arguments are being developed, they are arranged vertically. The flowchart holds much
more information.
Different Folks –People think and learn in different ways. When teaching, it is best to provide a few different
explanations and have a variety of ways to present content. Some students, the linear thinkers, will understand two-
column proof perfectly, and others, the special thinkers, will find flowchart proofs clearer. It is best to use both so
that all students understand and develop their reasoning skills. One option to introduce the flowchart format is to
have the students go back to key two-column proofs provided in the text and convert them to flowchart proofs.
Patterns and Structure –All of the shortcuts to triangle congruence require three pieces of information, therefore
the box of the flowchart proof that states that two triangles are congruent will have three boxes leading into it. These
kinds of structural relationships help students write and understand flowchart proofs and should be noted. It is also
helpful to give students incomplete flowchart proofs and have them fill in the missing information. Subsequent
proofs can be given with less information provided each time, until the boxes are all empty, and then with no help
at all. The only problem is that sometimes there is more than one way to write a proof and a different chart may be
required for the proof that the student wants to write. In that case, students can start from scratch if they like.
Proof Using SAS and HL
AAA –Students sometimes have to think for a bit to realize that AAA does not prove triangle congruence. Ask
them to think back to the definition of triangle. Congruent triangles have the same size and shape. Most students
intuitively see that AAA guarantees that the triangles will have the same shape. To see that triangles can have AAA
and be different sizes ask them to consider a triangle they are familiar with, the equiangular triangle. They can draw
an equiangular triangle on their paper, and you can draw an equiangular triangle on the board. The triangles have
AAA, but are definitely different in size. This is a counterexample to AAA congruence. Have the students note that
the triangles are the same shape; this relationship is called similar and will be studied in later chapters.
SSA –Student will have a hard time seeing the two possible triangles with SSA. The best way to describe it when
the congruent parts are set up, is to tell them to take that the last congruent side can bend in, so that the third side is
short to make one triangle, and bent out, so that the third side is long, to make the other triangle. Some students will
see it right away and others will really have to play around with their triangles for awhile in order to understand.
Why Not LL? –Some students may wonder why there is not a LL shortcut for the congruence of right triangles. It
Chapter 2. Geometry TE - Common Errors