Geometry>Quadrilaterals and Other Polygons>Polygon Interior and Exterior Angle Sums
Unit 1: Reasoning in Geometry
Day 1: Creating Definitions
Day 2: Inductive Reasoning
Day 3: Conditional Statements
Day 4: Quiz 1.1 to 1.3
Day 5: What is Deductive Reasoning?
Day 6: Using Deductive Reasoning
Day 7: Visual Reasoning
Day 8: Unit 1 Review
Day 9: Unit 1 Test
Unit 2: Building Blocks of Geometry
Day 1: Points, Lines, Segments, and Rays
Day 2: Coordinate Connection: Midpoint
Day 3: Naming and Classifying Angles
Day 4: Vertical Angles and Linear Pairs
Day 5: Quiz 2.1 to 2.4
Day 6: Angles on Parallel Lines
Day 7: Coordinate Connection: Parallel vs. Perpendicular
Day 8: Coordinate Connection: Parallel vs. Perpendicular
Day 9: Quiz 2.5 to 2.6
Day 10: Unit 2 Review
Day 11: Unit 2 Test
Unit 3: Congruence Transformations
Day 1: Introduction to Transformations
Day 2: Translations
Day 3: Reflections
Day 4: Rotations
Day 5: Quiz 3.1 to 3.4
Day 6: Compositions of Transformations
Day 7: Compositions of Transformations
Day 8: Definition of Congruence
Day 9: Coordinate Connection: Transformations of Equations
Day 10: Quiz 3.5 to 3.7
Day 11: Unit 3 Review
Day 12: Unit 3 Test
Unit 4: Triangles and Proof
Day 1: What Makes a Triangle?
Day 2: Triangle Properties
Day 3: Proving the Exterior Angle Conjecture
Day 4: Angle Side Relationships in Triangles
Day 5: Right Triangles & Pythagorean Theorem
Day 6: Coordinate Connection: Distance
Day 7: Review 4.1-4.6
Day 8: Quiz 4.1to 4.6
Day 9: Establishing Congruent Parts in Triangles
Day 10: Triangle Congruence Shortcuts
Day 11: More Triangle Congruence Shortcuts
Day 12: More Triangle Congruence Shortcuts
Day 13: Triangle Congruence Proofs
Day 14: Triangle Congruence Proofs
Day 15: Quiz 4.7 to 4.10
Day 16: Unit 4 Review
Day 17: Unit 4 Test
Unit 5: Quadrilaterals and Other Polygons
Day 1: Quadrilateral Hierarchy
Day 2: Proving Parallelogram Properties
Day 3: Properties of Special Parallelograms
Day 4: Coordinate Connection: Quadrilaterals on the Plane
Day 5: Review 5.1-5.4
Day 6: Quiz 5.1 to 5.4
Day 7: Areas of Quadrilaterals
Day 8: Polygon Interior and Exterior Angle Sums
Day 9: Regular Polygons and their Areas
Day 10: Quiz 5.5 to 5.7
Day 11: Unit 5 Review
Day 12: Unit 5 Test
Unit 6: Similarity
Day 1: Dilations, Scale Factor, and Similarity
Day 2: Coordinate Connection: Dilations on the Plane
Day 3: Proving Similar Figures
Day 4: Quiz 6.1 to 6.3
Day 5: Triangle Similarity Shortcuts
Day 6: Proportional Segments Between Parallel Lines
Day 7: Area and Perimeter of Similar Figures
Day 8: Quiz 6.4 to 6.6
Day 9: Unit 6 Review
Day 10: Unit 6 Test
Unit 7: Special Right Triangles & Trigonometry
Day 1: 45˚, 45˚, 90˚ Triangles
Day 2: 30˚, 60˚, 90˚ Triangles
Day 3: Trigonometric Ratios
Day 4: Using Trig Ratios to Solve for Missing Sides
Day 5: Review 7.1-7.4
Day 6: Quiz 7.1 to 7.4
Day 7: Inverse Trig Ratios
Day 8: Applications of Trigonometry
Day 9: Quiz 7.5 to 7.6
Day 10: Unit 7 Review
Day 11: Unit 7 Test
Unit 8: Circles
Day 1: Coordinate Connection: Equation of a Circle
Day 2: Circle Vocabulary
Day 3: Tangents to Circles
Day 4: Chords and Arcs
Day 5: Perpendicular Bisectors of Chords
Day 6: Inscribed Angles and Quadrilaterals
Day 7: Review 8.1-8.6
Day 8: Quiz 8.1 to 8.6
Day 9: Area and Circumference of a Circle
Day 10: Area of a Sector
Day 11: Arc Length
Day 12: Quiz 8.7 to 8.9
Day 13: Unit 8 Review
Day 14: Unit 8 Test
Unit 9: Surface Area and Volume
Day 1: Introducing Volume with Prisms and Cylinders
Day 2: Surface Area and Volume of Prisms and Cylinders
Day 3: Volume of Pyramids and Cones
Day 4: Surface Area of Pyramids and Cones
Day 5: Review 9.1-9.4
Day 6: Quiz 9.1 to 9.4
Day 7: Volume of Spheres
Day 8: Surface Area of Spheres
Day 9: Problem Solving with Volume
Day 10: Volume of Similar Solids
Day 11: Quiz 9.5 to 9.8
Day 12: Unit 9 Review
Day 13: Unit 9 Test
Unit 10: Statistics and Probability
Day 1: Categorical Data and Displays
Day 2: Measures of Center for Quantitative Data
Day 3: Measures of Spread for Quantitative Data
Day 4: Quiz Review (10.1 to 10.3)
Day 5: Quiz 10.1 to 10.3
Day 6: Scatterplots and Line of Best Fit
Day 7: Predictions and Residuals
Day 8: Models for Nonlinear Data
Day 9: Quiz Review (10.4 to 10.6)
Day 10: Quiz 10.4 to 10.6
Day 11: Probability Models and Rules
Day 12: Probability Using Two-Way Tables
Day 13: Probability Using Tree Diagrams
Day 14: Quiz Review (10.7 to 10.9)
Day 15: Quiz 10.7 to 10.9
Day 16: Random Sampling
Day 17: Margin of Error
Day 18: Observational Studies and Experiments
Day 19: Random Sample and Random Assignment
Day 20: Quiz Review (10.10 to 10.13)
Day 21: Quiz 10.10 to 10.13
Learning Targets
Generalize a formula for finding the interior angle sum of any polygon by decomposing the shape into triangles.
Given the number of sides of a polygon, determine the angle sum and solve for missing angles; given the angle sum, solve for the number of sides of a polygon.
Explain why the exterior angle sum of any polygon is always 360˚.
| Tasks/Activity | Time |
|---|---|
| Activity Questions 1-3 | 10 minutes |
| Debrief Questions 1-3 with Margin Notes | 5 minutes |
| Activity Questions 4-11 | 15 minutes |
| Debrief Questions 4-11 with Margin Notes | 5 minutes |
| QuickNotes | 5 minutes |
| Check Your Understanding | 10 minutes |
Activity: What's the Temperature in Here?
Lesson Handouts
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Answer Key
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Homework
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Our Teaching Philosophy:
Experience First,
Formalize Later (EFFL)
Learn More
Experience First
In the final two sections of this unit, we’ll turn our attention to other polygons. In this lesson, students begin by exploring the interior angle sum of triangles, quadrilaterals, and pentagons using a Geogebra applet. Students can change the vertices of the shapes and note that the interior sum stays the same. In questions 2 and 3 students explore why this is true. Instead of looking directly at the five interior angles of the pentagon, we look at the 9 angles created by dividing the pentagon into triangles. The sum of the nine angles is exactly the same as the sum of the five original angles! While this is not the only method for showing the interior angle sum, we find students to be very successful using this approach. Note that a diagonal is now defined as a segment connecting a vertex and a non-adjacent vertex, not just the opposite vertex, as is true for quadrilaterals. This is something to bring up in the debrief. Students should be able to make sense of the picture without using any formal definitions. After debriefing questions 1-3, let groups finish the rest of the activity through the end of page 2. The applet on question 4 is optional; many groups will be able to visualize the number of triangles in their head. Questions 5 and 6 are used to generalize an equation that relates the number of sides of a polygon to the number of triangles, to the total sum of the interior angles. On the back page, students look at exterior angles and recall from Unit 4 how these are created by extending each segment of a polygon in one direction. By noticing the five sets of linear pairs, students will see that the sum of the interior and exterior angles is 5(180) and the sum of the interior angles is 3(180), so the sum of just the exterior angles is 2(180) or 360˚. Finally, students consider what will happen when the number of sides changes.
Formalize Later
Today’s formalization will help students write the general equation for the interior angle sum of a polygon with n sides and generalize the argument for why the exterior angle sum is always 360˚. Before sending students to work on the Check Your Understanding questions, you may need to review the polygon prefixes of hexa-, penta-, hepta-, octa-, nona-, and deca-. Note that questions 3 and 5 require using both interior and exterior angle sums in tandem. Question 4 is a preview for tomorrow’s lesson when students study regular polygons in more detail.
