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# Proving Theorems About Triangles

Lesson 9 of 9

## Objective: SWBAT explain the proof of the Triangle Interior Angle Sum Theorem and the Triangle Exterior Angle Theorem.

## Big Idea: Proof Power! In this lesson, students the theorems we've proven make it possible to prove a host of other theorems.

#### Discovery

*20 min*

This lesson starts in the computer lab. I give students one simple instruction: Use Geometer's Sketchpad to make a conjecture about the sum of the interior angle measures of a triangle.

Within the allotted time, students should be able to show on Sketchpad that the sum of the interior angle measures of the triangle has a constant value of 180 degrees.

I walk around from student to student to make sure that every student has created the correct setup on Sketchpad for making the conjecture.

After 20 minutes, everyone should be finished and we head back to class.

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#### Guided Discovery

*15 min*

When we arrive back at class, I use another demonstration to show that the interior angle measures of a triangle sum to 180 degrees. It's the one where students cut the three angles off the edge of a triangle they've drawn and re-assemble these angles to form a straight angle measuring 180 degrees.

The point here is to give various types of evidence to make students believe the sum of the interior angle measures of a triangle is actually 180 degrees.

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#### Concept Development

*30 min*

Having convinced ourselves that the sum of the interior angle measures of a triangle is 180 degrees,in this section we go about formally proving this fact in order to establish it as a theorem that we can use in the axiomatic system we are building in this course.

So I model the proof of the Triangle Interior Angle Sum Theorem. See video example below.

After I model that proof, I move on to another proof: The Triangle Exterior Angle Theorem, which uses the theorem we just proved as well as the linear pair postulate, transitive property, and substitution. I'm hoping that students are starting to see how all of this really functions as a system.

In any case, I don't write the formal proof for the Triangle Exterior Angle Theorem. I only lay out the strategy of the proof. See first part of video below for and example of the strategy discussion. Students will take this strategy and write their own proof of the theorem as the homework for this lesson.

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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