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# Deductive Reasoning and Proof

Lesson 3 of 9

## Objective: SWBAT explain the proof of the vertical angles theorem

## Big Idea: True today, true tomorrow....but ALWAYS true? In this lesson, students learn how deductive reasoning is used to prove conjectures.

*80 minutes*

#### Activating Prior Knowledge

*10 min*

**Where We've Been:** We have just finished examining various cases of linear pairs and vertical angles. Students have made conjectures that linear pairs are always supplementary and vertical angles are always congruent. We also have an ongoing conversation about how inductive and deductive reasoning complement each other.

**Where We're Going:** We will be using deductive reasoning to write a proof of the vertical angles theorem. We will also do some basic problem solving that involves the linear pair postulate and vertical angle theorem.

In this section I basically remind students of what we've been doing. I say something like, "As you recall, yesterday we created linear pairs and vertical angles on Sketchpad and we noticed that linear pairs always seem to be supplementary and vertical angles always seem to be congruent. Today we're going to formalize these relationships. We're going to introduce the linear pair postulate and we are going to prove the vertical angles theorem.

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#### Concept Development

*30 min*

This is the first time we are really getting into deductive reasoning. First I give a definition: Deductive reasoning is the process of using logic to reach conclusions from given facts, definitions and properties. Facts can be further broken down into two categories. We have postulates which are accepted as fact without proof and we have theorems which are established as fact through proof.

There are various definitions that we can use in proofs. Too many to list.

And then there are properties. For the most part, we will be using two types of properties. The first type, properties of equality, includes:

- Addition property
- Subtraction property
- Multiplication property
- Division property
- Substitution property
- Transitive property
- Reflexive property
- Symmetric property

I give the symbolic language for each of these properties and also an example. Students record these in their notes.

The second type of property, properties of congruence, includes:

- Reflexive property
- Symmetric property
- Transitive property

Again, I give the symbolic language for each property along with an example. Students record these in their notes.

Next I check for understanding using whiteboards.

The last thing to do before starting on the proof of the vertical angles theorem is to introduce the linear pair postulate. Although this is sometimes treated as a theorem, it is pretty self-evident. So I choose, as many texts do, to treat it as a postulate.

Then I go on to proving the vertical angles theorem. It is a rich proof that also has the proof of the congruent supplements theorem embedded in it. See [educreation] for a rendition of the proof.

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#### Guided Practice

*20 min*

While students won't be writing their own proofs at this point, I start to get them used to the idea of justifying steps in problem solving using facts, definitions, and properties...ok I guess they will be writing their own proofs.

I model this for students before asking them to do it themselves. I use the Linear Pair and Vertical Angles Problem Solving with Justification[BlankModeled] resource to show students what we're aiming for. I start with the blank version of the handout (Linear Pair and Vertical Angles Problem Solving with Justification[Blank].pdf) and fill it in on the document camera, explaining why I'm doing what I'm doing as I do it.

I leave this as a worked example for students to reference as they solve similar problems in the next section of the lesson.

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#### Independent Practice

*20 min*

Students complete the Linear Pairs and Vertical Angles Problem Solving with Justification [Independent] resource unassisted. As students bring me the assignment to turn it in, I glance over it. If there are any mistakes, or if the work flow or explanation is not sufficient, I give it back to the student to revise before resubmitting.

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras

- LESSON 1: Vertical Angles and Linear Pairs
- LESSON 2: Conjecture is Not Enough: The need for proof
- LESSON 3: Deductive Reasoning and Proof
- LESSON 4: Developing Two-Column Proof Skills
- LESSON 5: Exploring Parallel Lines Cut By a Transversal
- LESSON 6: Applying Postulates and Theorems Involving Parallel Lines Cut by a Transversal
- LESSON 7: Proving Theorems involving Parallel Lines Cut by a Transversal
- LESSON 8: Making Conjectures about the Midsegments of a Triangle
- LESSON 9: Proving Theorems About Triangles