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# Triangle Sum Theorem Proof

Lesson 10 of 16

## Objective: SWBAT use their observations to deduce the Triangle Sum Theorem.

#### Paper Corners Review

*15 min*

I created this lesson as an alternate approach to proving triangle angle sums. I find that many students are already familiar with the activity in which they cut off the angles and line them up to form a straight line. So instead of repeating the experiment, I start by reviewing some homework problems and then I show the class this clear although somewhat bizarre video:

Source: http://youtu.be/BQsO4WxAVb8

After the video I start a discussion around the concept of proof, "Could we use this experiment to show that this works for every type of triangle? Would that be possible?"

Here I want students to deduce that we need a better way to approach the problem. With a proof that looks at specific cases, we will never be able to prove that it always works for every case of a triangle, since we can always make a *new *case slightly different from the one before. I conclude by announcing, "Today you are going to prove that *all* triangles have 180 degrees. And you are going to do it without cutting up any triangles."

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#### The Geogebra Module

*20 min*

I have created a Geogebra Module for my students that represents a basic variation of the triangle sum proof:

http://www.geogebratube.org/student/m80592

My goal for my students is that they can build on our work from yesterday's class to create a proof of the Triangle Angle Sum theorem. The demonstration in the module by encourages students to use what they already know about alternate interior and supplementary angles.

I give them Module 6 Guide as a source of prompts to guide their exploration. A key to supporting the success of this lesson is to allow students to find the words that describe the angle relationship and support the writing of a proof.

#### Resources

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#### Summary and Extension

*25 min*

In this part of the lesson, we ask several students to share their versions of their Triangle Angle Sum proof. I will show show the Powerpoint slides (Transversals) that include the Geogebra module again to spark conversation. I will ask students to volunteer to present their findings. I like to quote students when they have written particularly effective arguments. I will often put several proofs on the board during this lesson.

At this point many students still use specific number cases to make their argument, but I challenge them to think about the different between using specific angle measurements and variable angle measurements.

We finish today with about 10 minutes on these extension problems:

Before departing, we review students' answers and discuss the meaning of having similar or congruent triangles (these words appear in the last problem).

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- LESSON 1: Developing Right and Straight Angle Intuition
- LESSON 2: Create Problems with Right and Straight angles
- LESSON 3: Why Are Vertical Angles Equal?
- LESSON 4: Create Vertical Angle Problems
- LESSON 5: Developing Transversal Intuition
- LESSON 6: Create Transversal Problems
- LESSON 7: Why Do Triangles Have 180 Degrees?
- LESSON 8: Walking Around a Triangle
- LESSON 9: Defining Key Angle Relationships
- LESSON 10: Triangle Sum Theorem Proof
- LESSON 11: Angles and Algebra
- LESSON 12: Super Practice with Angle Values
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