Proving Theorems involving Parallel Lines Cut by a Transversal
Lesson 7 of 9
Objective: SWBAT informally explain the proofs of theorems involving parallel lines cut by a transversal.
Activate Prior Knowledge
Where We've Been: We've just finished making conjectures about angle pairs formed when parallel lines are cut by a transversal.
Where We're Going: Students will eventually write proofs of the theorems for these angle pair relationships.
In this lesson, I want students to walk away with a conceptual understanding of the proofs, even if they are not able to write the proofs on their own.
The proofs we'll be writing involve the following content we have already learned: vertical angles theorem, linear pair postulate, congruent and supplementary angles, transitive property, substitution property, subtraction property. So the aim of this section of the lesson is to make sure that "systems are go" with all of this prior knowledge.
As a class, we complete the PLCT Proofs[APK] resource. Starting with #1, I ask students to think, reference their notes, etc. to come up with a verbal and visual representation of the vertical angles theorem. We then do a pair and a share. Finally, I model the desired final product on the document camera.
When I'm satisfied that students have these prerequisites down, I get into the lesson.
The focus of this lesson is obviously proving theorems involving parallel lines cut by a transversal, but the lesson is also part of a learning progression related to axiomatic systems. The goal in this section of the lesson is to be explicit about what an axiomatic system is and how axiomatic systems operate.
Axioms, or postulates, are the statements that we decide (or agree) to accept as true and self-evident without proof. Postulates enable us to prove theorems, which can then be used to prove other theorems. I remind the students how we used the linear pair postulate to prove the vertical angles theorem, which we will (by the way) be using to prove theorems in this lesson.
In the present lesson, I relate to students, we start with the corresponding angles postulate. In other words, we accept without proof that when parallel lines are cut by a transversal, all pairs of corresponding angles will be congruent. This postulate will allow us to prove other theorems about parallel lines cut by a transversal. These new theorems, in turn, will allow us to prove more theorems (e.g. the Triangle Interior Angle Sum Theorem). As I discuss these ideas conversationally with students, I also condense the main points into notes that they can keep for their records.
In this section of the lesson I am doing two things:
- Documenting the proofs for students so that they can refer to them later
- Modeling the strategies I use when I write proofs
While it is certainly important for students to have a record of the proofs, they can easily get this from a geometry text or some other reference document. The strategies used to produce the proof, though, are expert knowledge that needs to be carefully conveyed...by an expert.
Here are some of the strategies that I model:
- Think Plot Before Dialogue: I have a hunch that authors and screenwriters have a good idea of their plot before they start writing dialogue. Similarly, when I write a proof, I have a basic blueprint of the proof before I start to add all of the rigor and detail. This can be done verbally or visually [see educreation]
- Cross the Creek: When crossing a creek, we tend to find a series of stable rocks that are close enough to each other and will lead us from one side to the other. Similarly, when writing proofs, we have to find a series of statements and reasons, one leading to the next that get us from our givens to whatever we're trying to prove. Skip a step, and you fall in the water.
- Keep Your Eye on the Prize...and the Gap: Proofs are all about sustaining focus on what we're trying to prove and how that relates to our current position in the proof. We are continuously trying to close that gap in the most efficient way possible.
- Talk to Yourself: As I am writing a proof, I ask myself questions like "How do I know that?" "What allows me to say that?" "Now that I've established that, what am I able to say now?" "How does that help me to prove what I'm trying to prove?" "How does this statement follow from the previous statement(s)?"
So the way this section of the lesson goes is I carefully model the following two proofs:
- Alternate Interior Angles Theorem
- Consecutive (Same-Side) Interior Angles Theorem
As I'm modeling these proofs (and strategies), I make students put their pencils and pens down to make sure that their full attention is devoted to understanding the proofs. I give them time to copy the proofs when I am done.
Once I've modeled these two proofs and done some basic checking for understanding to make sure that the majority of students are grasping the concepts, I move on to two analogous proofs:
- Alternate Exterior Angles Theorem
- Consecutive Exterior Angles Theorem
With these two proofs, I gradually release control to the students. This is a way for me to see if they have truly understood the first two proofs I modeled, and it is an opportunity for students to develop their skill and self-efficacy writing proofs.
I start by asking (for each proof), what our basic plot is going to be. I do this through a think-pair-share so that everyone has a chance to grapple with it. Once we agree on our overall plan (the bare bones) for the proof, I take volunteers to try their hand at fleshing out the steps of the proof. Of course, I'm there to get us back on track when we go astray. Typical missteps include, making extraneous statements or attempting to make statements that have no basis yet in the proof. At the end of this process, I again give students a chance to copy the final versions of the proofs.
The independent practice for this lesson is a take-home assignment. Students are required to take two of the theorems we proved in the lesson (one for alternate angles and one for consecutive angles) and write a paragraph proof for each. I have already modeled paragraph proofs during an earlier lesson on proofs.