Congruence in Two Dimensions

Lesson 3

Math

Unit 2

10th Grade

Lesson 3 of 18

Objective


Describe and apply the sum of interior and exterior angles of polygons.

Common Core Standards


Core Standards

  • G.CO.C.11 — Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Foundational Standards

  • 8.G.A.5

Criteria for Success


  1. Use the sum of the interior angles of a triangle to derive the sum of the interior angles of any polygon. 
  2. Describe the relationship between the number of sides and the sum of the interior angle measures of a polygon. 
  3. Use supplementary angle relationships to determine individual exterior angle measures of polygons.
  4. Describe why the sum of the exterior angles of any polygon sums to 360. 
  5. Use the interior and exterior angles of polygons to find missing measures.

Tips for Teachers


This lesson has been written to meet the Massachusetts standard of G-CO.11a, specifically addressing interior and exterior angles. The Common Core standard G-CO.11 does not specifically call out interior and exterior angles.

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Anchor Problems


Problem 1

Which of the following figures has the largest sum of interior angles?

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

Dan Meyer Recipes for Surprising Mathematics

Recipes for Surprising Mathematics by Dan Meyer is made available at dy/dan under the CC BY 4.0 license. Accessed March 8, 2017, 11:47 a.m..

Modified by Fishtank Learning, Inc.

Problem 2

Which of the following has the greatest sum of exterior angles?

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

Easing the Hurry Syndrome Angles in Polygons Problems - Items - Tasks

Angles in Polygons Problems - Items - Tasks by Jennifer Wilson is made available on Easing the Hurry Syndrome. Accessed Aug. 10, 2017, 10:02 a.m..

Target Task


Problem 1

Compare the quantity in Column A with the quantity in Column B. 

Column A Column B
The measure of each exterior angle of a regular triangle The measure of each exterior angle of a regular hexagon
  1. The quantity in Column A is greater. 
  2. The quantity in Column B is greater. 
  3. The quantities are equal. 
  4. The relationship cannot be determined on the basis of the information given. 

References

JMAP Geometry Resources by Standard

Geometry Resources by Standard is made available on JMAP by Steve Sibol and Steve Watson. Copyright © 2017  JMAP, Inc. - All rights reserved. Accessed Aug. 10, 2017, 10:07 a.m..

Modified by Fishtank Learning, Inc.

Problem 2

Find the measure of an interior angle and an exterior angle of a regular polygon with 40 sides.

References

JMAP Geometry Resources by Standard

Geometry Resources by Standard is made available on JMAP by Steve Sibol and Steve Watson. Copyright © 2017  JMAP, Inc. - All rights reserved. Accessed Aug. 10, 2017, 10:07 a.m..

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Use the idea of tessellation and the 360 degrees of a circle to predict which numbers of sides would fit “snugly” together. See Sam J. Shah's article Snug Angles.
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Lesson 2

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Lesson 4

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Introduction to Polygons

Topic B: Rigid Motion Congruence of Two-Dimensional Figures

Topic C: Triangle Congruence

Topic D: Parallelogram Properties from Triangle Congruence

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