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# Interior and Exterior Angle Sum of Polygons

Lesson 2 of 9

## Objective: SWBAT apply their knowledge of the interior and exterior angle sum measures for polygons to solve for indicated angles.

## Big Idea: Through investigating a variety of polygons with different numbers of sides, students will discover the interior and exterior angle sum for any polygon.

*92 minutes*

In the Regular Hexagon Exploration, students use a compass and straightedge to construct a regular hexagon, which gives them a kinesthetic way to start making sense of the structure of a regular hexagon as they discover its properties. Students work in groups to find angle measures and justify their reasoning using definitions and theorems. The task is open-ended. I ask groups to come up with their own conjectures about regular hexagons based on their construction. These conjectures, for example, can be about angle measures or diagonals. The goal is for my students to convince themselves, to convince a friend, and to convince a skeptic. I take this approach to enable their spirit of discovery while maintaining rigor (**MP3**).

To ensure students write high-quality justifications, I give students an example **sentence frame** and include a **word bank** in the task (vertex, diagonal, adjacent, perpendicular, right angle, bisect, midpoint, congruent, CPCTC). For example:

**Given a regular hexagon, there is a diagonal that divides the regular hexagon into two congruent trapezoids. I know this because…**

As student work I take notes on how they are working in groups to encourage the kind of math talk I want in my classroom.

We debrief the exploration on the whiteboard, with students sharing out some of the conjectures they came up with.

I would like to thank Allison Krasnow, a math/technology coach in Berkeley who share this hexagon diagram with me.

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The goal of the Polygon Interior Angle Sum Conjecture activity is for students to conjecture about the interior angle sum of any n-gon. Students will see that they can use diagonals to divide an *n*-sided polygon into (*n*-2) triangles and use the triangle sum theorem to justify why the interior angle sum is (n-2)(180). They will also make connections to an alternative way to determine the interior angle sum, noticing that (*n*-2)180 =* *n(180)-360. The activity provides a nice change of pace from hands-on construction to hands on analysis, looking for patterns. I often find that changing pace like this helps me to better engage all of my students in learning.

#### Resources

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Students work in groups in the Polygon Exterior Angle Sum Conjecture activity to conjecture about the exterior angle sum of any n-gon. Each student in the group is given a different polygon with its exterior angles drawn out. After each member of the group traces all of the exterior angles so that they are adjacent, the group compares their results to conjecture about the exterior angle sum for any polygon. The group must then prove the exterior angle sum for a triangle, quadrilateral, and then any n-gon before checking in with the teacher.

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#### Debrief Notes

*15 min*

Like most of our lessons, we debrief the big ideas for the day in our note takers. We debrief students’ conjectures about the interior angle sum of polygons and extend this idea further to determine the measure of each interior angle of any equiangular n-gon. Then we debrief students’ exterior angle sum conjectures—during this part of the debrief, I like to have at least one group of students share out their proofs with the whole class.

#### Resources

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#### Homework

*5 min*

The Polygon Sum and Exterior Angle Sum Theorem Homework for this evening offers my students some challenge as they review the work that was completed in class today.

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- UNIT 1: Creating Classroom Culture to Develop the Math Practices
- UNIT 2: Introducing Geometry
- UNIT 3: Transformations
- UNIT 4: Discovering and Proving Angle Relationships
- UNIT 5: Constructions
- UNIT 6: Midterm Exam Review
- UNIT 7: Discovering and Proving Triangle Properties
- UNIT 8: Discovering and Proving Polygon Properties
- UNIT 9: Discovering and Proving Circles Properties
- UNIT 10: Geometric Measurement and Dimension
- UNIT 11: The Pythagorean Theorem
- UNIT 12: Triangle Similarity and Trigonometric Ratios
- UNIT 13: Final Exam Review

- LESSON 1: Review Angle Relationships to Begin Conjecturing About Polygons
- LESSON 2: Interior and Exterior Angle Sum of Polygons
- LESSON 3: Going Deeper with Interior and Exterior Angles
- LESSON 4: Focus on Justification: Interior and Exterior Angles
- LESSON 5: Kite & Trapezoid Properties and Midsegments of Triangles & Trapezoids
- LESSON 6: Properties of Parallelograms and Special Parallelograms
- LESSON 7: Proofs: Properties of Parallelograms
- LESSON 8: Review: Polygon Properties
- LESSON 9: Discovering and Proving Polygon Properties Unit Assessment