Proving Triangle Congruence Shortcuts with Transformations
Lesson 4 of 4
8th or HS Geometry
100-180 minutes
Description
Through a "Simon Says" style game, students prove triangle congruence theorems SSS, SAS and ASA using transformations.
Materials
Introduce
Review the three proposed criteria for triangle congruence. Make sure the statements are clear and precise (SMP 6). Remind students that we haven't proved them yet, just conjectured.
Listen to students' ideas about what it would take to prove they work for any triangle. Summarize student statements with something like the following: "We'd have to prove that no matter where the triangles are or what the sides and angles are, as long as one of the sets of criteria is true, we can definitely always translate, rotate, and reflect to get one triangle onto the other."
Explore
Part 1: Simon Says Proofs
Introduce students to the Simon Says Proofs directions (slide 3 of the
PowerPoint slides), where the teacher is Simon and the students are the players.
Encourage students to try each of the starting conditions several times. Each time a new student takes the role of Simon.
Bring students back together. Ask them the following questions:
- "Were there any starting conditions where the triangles were NEVER congruent?"
- "Were there any starting conditions where the triangles were SOMETIMES congruent?"
Emphasize again that, "We want to prove that no matter where the triangles are or what the sides and angles are, as long as one of the sets of criteria is true, we can definitely always translate, rotate, and reflect to get one triangle onto the other."
Ask, "How does getting good at the Simon Says game help us show that, no matter where the triangles are or what the sides and angles are, as long as one of the sets of criteria is true, we can definitely always translate, rotate, and reflect to get one triangle onto the other?"
Have each group of four play the game until they are confident that they could come up with a set of transformations that map one triangle onto the other no matter where the triangles begin. Have students fill out a flowchart for one or more starting conditions. Suggestion: Students might do this as homework.
Suggestion for Differentiation:
- ASA: This proof is probably the simplest, as the justification for each step follows from what we know to be true.
- SAS: This proof is a bit harder to justify the key steps.
- SSS: This proof is the hardest to justify the key steps.
- Suggestion for working on SMP 3: Have each pair/group swap flowcharts with another pair (perhaps a pair that had different criteria), check each other's chart for clarity and accuracy on a new pair of triangles, and then compare to find similarities and differences.
Part 2: Building the Proofs
Collect students' flowchart descriptions of how they would transform an arbitrary triangle satisfying the given conditions onto another arbitrary triangle.
Choose one or two steps of the flowchart to ask students to use the definitions of rigid motions to prove how they know that Point E will end up mapped onto Point B, and/or Point F will end up mapped onto Point C. Most students will use a translation, followed by rotation, followed by reflection (if needed). In that case:
- For students' rotation step, ask, "How do you know that if Ray D'E' coincides with Ray AB, that Point E' coincides with Point B?"
- For students' reflection step, ask, "How do you know that, after reflection, Ray AC will correspond with Ray D''F'' and C and F'' will correspond?"
This should result in a somewhat formal proof by transformations for each of the three triangle congruence cases.
Here are fully justified responses for the SSS, ASA, and SAS cases. Note that for the AAS case we can calculate that the 3rd pair of angles must be congruent and turn the problem into an AASA case which works just like ASA.
ASA:
- Translate so that D' coincides with A. Now (at least) one pair of points coincides.
- Rotate around A so that Ray D''E'' coincides with Ray AB. We know that E'' will correspond with B because rotations preserve distance and it's given that AB = DE. If B is the same length from A that E'' is from D'', and they are on the same ray, then they coincide.
- Either all three points will now coincide and we're done, or we need to reflect over Segment AB.
- After reflection (if needed), all three points will coincide, because reflecting over AB won't move D''' or E'''. After reflection, Ray D'''F''' will coincide with Ray AC and Ray E'''F''' will coincide with Ray BC because reflections preserve angles, and rays coming from the same point at the same angle coincide. Ray AC and Ray BC can only intersect at one point, C, and Ray D'''F''' and Ray E'''F''' can only intersect at one point, F''; since the rays coincide, their intersection points, C and F''', must, too.
- Therefore, all three points will definitely coincide after translation, rotation, and reflection (if needed).
SAS:
- Translate so that D' coincides with A. Now (at least) one pair of points coincides.
- Rotate around A so that Ray D'E' coincides with Ray AB. We know that E' will correspond with B because rotations preserve distance, and it's given that AB = DE. If B is the same length from A that E' is from D', and if they are on the same ray, then they coincide.
- Either all three points will now coincide and we're done, or we need to reflect over Segment AB.
- After reflection (if needed), all three points will coincide because reflecting over AB won't move D'' or E''. After reflection, Ray D'''F''' will coincide with Ray AC because reflections preserve angle measure and we are given that Angle A = Angle D. When Ray E''F'' coincides with Ray BC, then F'' and C have to coincide, because reflections preserve distance and it's given that AC = DF. If C is the same length from A that F''' is from D''', and they are on the same ray, then C and F''' coincide.
- Therefore, all three points will definitely coincide after translation, rotation, and reflection (if needed).
SSS:
- Translate so that D' coincides with A. Now (at least) one pair of points coincides.
- Rotate around A so that Ray D''E'' coincides with Ray AB. We know that E'' will coincide with B because rotations preserve distance and it's given that AB = DE. If B is the same length from A that E'' is from D'', and if they are on the same ray, then they coincide.
- Either all three points will now coincide and we're done, or we need to reflect over Segment AB.
- After reflection (if needed), all three points will coincide, because reflecting over AB won't move D'' or E'' off of A and B. Before reflecting, the situation will look something like this (recall that A and D'' coincide, and B and E'' coincide):
- We know that AC = D''F'' (given) and BC = E''F'' (given). Therefore, since D'' coincides with A and E'' coincides with B, A and B are both on the perpendicular bisector of Segment CF''. Since we are reflecting over AB, we're reflecting over the perpendicular bisector of C and F''. When you reflect over the perpendicular bisector of two points, the points are mapped onto one another, so F''' will coincide with C after reflection.
- Therefore, all three points will definitely coincide after translation, rotation, and reflection (if needed).
Synthesize
The authors wish to acknowledge that students writing these transformation proofs with some degree of rigor may feel that they should be treated to a parade led by the superintendent, and get a key to the city from the mayor or something. These proofs are HARD in that they stretch students (and their teachers) to think about Geometry differently. But, these proofs are doable and a sign that students have mastered productive struggle! What can you do with your students to celebrate their success?
One important aspect of summarizing is to debrief about what helped kids persevere with the Simon Says game and the proofs that their flowchart steps work as well as how awesome it was that they did persevere and how proud they can be of what they came up with.
The other point to emphasize in this Lesson ARC is how these proofs help them understand which triangles are good candidates for being congruent and how much simpler it is to use these proven shortcuts (SSS, ASA, SAS) vs. having to come up with a general mapping rule for every new situation they encounter. So, for example, instead of coming up with transformation proofs for each new triangle situation they encounter, they can just figure out how to "turn these into a problem I've solved before" by figuring out which congruence theorem shortcuts are relevant in each case (SMP 7).
Teacher Reflection
Build procedural fluency from conceptual understanding:
- To what extent were the students able to make sense of and articulate why the Simon Says game helps us prove that these triangle shortcuts always work?
Pose purposeful questions:
- Which questions about students' flowcharts led to reasoning about the properties of transformation and how they preserved distance?
Support productive struggle in learning mathematics:
- What more would students have to add to their flowcharts-with-justification to have a rigorous proof?
Leave your thoughts in the comments below.