Rigid Motion, Congruent Triangles, and Proof
Lesson 1 of 6
Objective: SWBAT identify the reasons that triangles are congruent, using SAS, ASA, SSS, and AAS, by closely examining diagrams and "givens."
To begin this Unit (and this lesson), I let the students know that we are going to use our knowledge of rigid motion, gained in the previous chapter, to write proofs that two triangles are congruent. As an opening, I plan to discuss the meaning of the word congruent, and, how rigid motions might be used to show congruence. We will also discuss the meaning of congruence with respect to the parts of a triangle.
I plan to take things as far as explaining that we are looking for a series of isometries that will map each side and each angle of one triangle onto another. I'll say, "If we can find these isometries, we will have proven that the triangles are congruent. Another way of saying this is that one triangle is an image of the other."
My students often want to know why we are going to do this. If they ask, I'll say, "In this unit, we are trying to determine that minimum amount of information that we need to know in order to prove congruence." They are usually satisfied by this response.
Our first case will be that of Side-Angle-Side. To illustrate this, I am going to use a YouTube video. This video is silent, which I like because this allows me to explain as the students watch:
Source URL: https://www.youtube.com/embed/30dOn3QARVU (accessed Nov 18, 2014)
I introduce the other theorems - ASA, SSS, and AAS - and we continue to discuss these in terms of rigid motion.
To begin, we will practice identifying why two triangles are congruent. For this task, I will use a worksheet that I found online, provided by Kuta Software. I have found many of Kuta worksheets to be helpful when I am looking for a resource to help my students practice a particular concept.
I will do the first two or three problems with the students. Then, I will ask them to complete the worksheet in their groups, discussing the concepts as they go. When most students are finished, I plan to call on students to read their answers to a problem, working my way around the room systematically. When there is disagreement, I will ask students to explain their reasoning, beginning with the original respondent.
As we discuss answers, I will stress the importance of marking the diagrams. I want to make sure that my students recognize the importance of this step in solving a problem. It is difficult to discuss triangle congruence if congruent parts are not clearly and accurately indicated in the diagram (MP6). A "rule" that I stress is that three sets of congruent parts must be marked each time in order to prove that two triangles are congruent. I have found that this is not always obvious to students; they will work with triangles that appear to be congruent without confirming their observation. The theme of carefully and fully annotating diagrams will be present throughout this unit.
Let's Try Some Proofs
I have copied the document entitled Statements Used in Geometric Proofs onto brightly colored paper. (I have found that using colored paper for important information helps my students to find these sheets quickly and drives home the fact that this is important information.) Much of the information on this sheet is familiar to my students. By placing it all on one sheet, however, the students are able to easily refresh their memories and remind themselves of their possible options.
After I hand out Statements Used in Geometric Proofs, I give the class some time to look the statements over. I make sure that students locate the ways of proving triangles listed on the back. Then, I hand out the Beginning Proofs.
I have chosen to present these opening proofs as flow chart proofs. I have been using flow chart proofs in class for several years now as an introduction, and have found that they are a great way to start. Each year I have experimented with them a little more. Now, I am pretty pleased with how my students develop their understanding from these visual proofs.
For today, each proof has been laid out for the students so that it is clear that they need to find three pairs of congruent triangle parts, and, to do so, they must focus on each "given," one at a time. When they have used up all the of given statements, but still need to prove another set of triangle parts are congruent, I emphasize that they need to look closely at the diagram to determine if any other information is contained in the diagram (e.g., vertical angles or a reflexive side).
We'll work on the first proof as a class. As we do I will emphasize the importance of reading and writing down each given, and marking the diagram. (I provide colored pencils (MP5) for the marking of the diagrams.) When I feel that the class has a good understanding of the first proof, I ask them to work on the second proof, and we take our time discussing and answering questions on it, proceeding in this manner, through all four proofs on the worksheet.