An Introduction to Proof and Parallel Lines

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SWBAT identify the angle pairs created by parallel lines, and solve for angle measures. SWBAT solve for angle measures using the vertical angles theorem and apply the theorem to proofs.

Big Idea

Students are introduced to the two-column proof, and put this knowledge to work on vertical angles and the angle pairs created by parallel lines and transversals.

Handing Back the Unit Test

15 minutes

I am a firm believer in handing tests back during the following lesson.  Sometimes it is challenging to get tests corrected so quickly, but I think it's important to provide prompt feedback to the students.

I hand the first unit test back and provide students time to talk within their groups about their answers.  I then ask the students if there are any questions that they would like me to answer.  For this particular test, rather than my working out the two coordinate geometry problems out for students, I ask two or three students who did a particularly nice job on them to display their solutions on the board using a document camera.  I explain in detail exactly what I am looking for in terms of justification and explanation, and let all of the students know that they have the opportunity to re-do these two questions (again and again, if need be, until they've done them well!) and to improve their grades by doing so.  I believe that these two problems set the tone for the year, in terms of my expectations for justifications and the quality of their work.

Properties from Algebra and the Vertical Angles Proof

15 minutes

In this unit on parallel lines, I introduce students to proof, using six algebraic properties: the reflexive, substitution, addition, subtraction, multiplication, and division properties.  For the proofs that we will be doing in this lesson, I use the two-column format specifically because its vertical arrangement is very similar to the arrangement of the way I solve algebraic problems and these proofs are pretty algebraic in nature.  Later in the course I will provide the students with experiences using flow chart proofs and paragraph proofs; in this case, however, the two-column proof makes the most sense to me.

I provide the students with an algebraic properties handout.  I give the students time to read the properties, ask if there are any questions, and then give them time to fill in the properties that are used in the two algebraic problems.  The first step of the first example (3 = 3) will seem kind of random to the students, but I take the time to explain that in Geometry, when we use a property that refers to "equal quantities" being added or subtracted, we will actually need to state that the quantities are equal by using the reflexive property.  So, while it seems weird right now to make such a statement, this property will be used geometrically later in the lesson.

When everyone has finished filling out the properties, we discuss them as a class.  I stress that it is just as important when we are working with angle or segment measures in Geometry to keep our equations balanced as it is in Algebra.

The first geometric theorem that we prove is Vertical angles are congruent.  We will need this theorem to prove two of our parallel line theorems.

I provide the students with the vertical angles handout and we work together through this proof.  I begin by asking What can you tell me about angle AOB and angle COD? How do you know this? I continue in this vein - what do you know? how do you know this? - recording their statements and reasons on the board and asking that they fill in their sheets as we progress. I try to model the thought process that they will need to master in order to do geometric proofs. 

When we have finished the proof, I inform the class that we can now use this theorem as a reason in our future proofs. For what its worth, I always mention that when I was in high school a million years ago, we called these angles opposite angles. To me, this makes a lot more sense than the name vertical angles, as they are indeed angles that are opposite one another. Does anyone know why the math gurus changed the name? I wish they had run that change by me!

[I have provided my method for doing this proof here.  I realize that it can be simplified, or that it can be made even more complicated.  (However, I believe that more complicated does not necessarily equate with more rigor!)  I have chosen this particular approach because it reinforces a lot of what we have been working on in this and the previous unit.]


The Angles Formed When Any Two Lines Are Cut by a Transversal

20 minutes

I now explain that we are going to shift from vertical angles to the angles formed by two lines that are cut by a transversal.  I know that the students have studied these angles before, in eighth grade, but I have found in the past that it is still necessary to "hammer" the meaning of the vocabulary, including the words interior, exterior, alternate, and same-side.  (See my video for my views on this!)

After the students and I discuss the relevant vocabulary and look at a few examples on the board, I hand out and explain our next activity, a "foldable".  Because so many students have struggled with this topic in the past, I have decided to give this activity a try, hoping it will help to make the topic clearer for those who struggle.  Students will need colored pencils or crayons and scissors to complete it. 

I hand out and explain the foldable.  I also pass around a sample that I have completed, for any students who are having trouble understanding the assignment.  Students work together in their groups to complete the work.

[Here's another helpful image. I have totally and completely borrowed this activity from the blog of a wonderfully creative eighth grade teacher named Kathryn. (Thank you, Kathryn!) If you go to her website, you will find even more photos and a full explanation.]

At this point, I have not yet mentioned parallel lines.  The angles that we are talking about in this section are angles formed by any two lines cut by a transversal.  I have observed way too often that some students think all corresponding angles are congruent or that all same-side interior angles are supplementary, completely missing the fact that these statements are only true when two lines are parallel.  When we get to writing reasons in proofs involving parallel lines, students will repeatedly ask if they can't just write, "Alternate interior angles are congruent" and my answer is always "No!" for just this reason.

Properties of Parallel Lines

35 minutes

In our first unit, the students were introduced to parallel lines, so I only briefly revisit the meaning of parallel, stressing the fact that parallel lines are coplanar, and then contrast them with the concept of skew lines.  

I then pose the question What is true about the angle pairs formed by two parallel lines cut by a transversal? I provide the students with a protractor (MP5), ask the students to draw two parallel lines and a transversal, measure the angles formed, and to develop their own hypotheses about the three types of angle pairs. (MP3)  We discuss their hypotheses, and then I explain that we will postulate the theorem if two parallel lines are cut by a transversal, the corresponding angles are congruent because we have not yet learned everything that we need to know to prove this. 

From this postulate, we work on proving the other two theorems.  I hand out the Parallel Lines Proofs to the students and I give them time to work on the proofs in their groups before we discuss them.  In our discussion, I am again careful to ask the questions What do you know from this statement? How do you know this?  I have entered some statements and reasons in the proofs, as an aide for the students.

When we have finished proving these theorems, I explain that we can now use these theorems as reasons in future proofs.

To cement their recognition of these angle pairs and their relationships, I use a set of powerpoint slides.  I have used these slides in the past and the students seem to really like doing them.  The slides are black, with florescent colors, so I turn off the lights in the classroom to increase their impact.  The problems are quick and easy - paper, pencils, and calculators are not needed - and provide a nice change of pace.  The slides focus on a variety of angle pairs (in addition to the angles created by parallel lines, there are vertical angles and linear pairs).  For each slide I ask the students to determine the type of angle pair, as well as the value of x.  In the last few slides, the students will need to do two steps to find the angle measure, and this provides opportunity for the students to discuss the various ways they can arrive at the value of x. (MP3)

Throughout this activity, I stress that students should be asking themselves Does this answer make sense? If one angle is obtuse and one angle is acute, can the angles possibly be congruent? Do the angles look congruent?  These questions may seem like duh questions, but too often students make mistakes simply because they have not asked themselves these types of questions. (MP1)

Closing and Homework

5 minutes

To close the lesson, I give a ticket out the door.  As the students hand in their ticket, I graciously exchange their ticket for the homework assignment.

The homework consists of 15 fairly straight-forward parallel line problems.  They are all multiple choice questions, with the diagrams provided, so that I'm hoping that the students will be able to complete the assignment without too much difficulty.  I am not a huge fan of multiple choice problems, but I know that I have to provide practice on them anyway, because they are such a big part of our state assessment.  

(These problems are directly off past New York State Regents tests.  If interested in seeing more Regents Geometry problems, they can be found at the New York State website and at jmap.)