Sum of Angles in Convex Polygons: An Interactive Guide

2/28/24, 11:23 AM Lesson Activity: Investigating the Sum of Angles in Convex Polygons https://f1.app.edmentum.com/courseware-delivery/ua/125276/45610409/aHR0cHM6Ly9mMS5hcHAuZWRtZW50dW0uY29tL2x0aS92Mi9sZWFybmV… 1/8 Lesson Activity Investigating the Sum of Angles in Convex Polygons This activity will help you meet this educational goal: You will find the sum of the interior angles of a convex polygon, determine a rule for finding the sum of the interior angles for all convex polygons, and understand why it works. Directions Read the instructions for this self-checked activity. Type in your response to each question, and check your answers. At the end of the activity, write a brief evaluation of your work. Activity In this activity, you'll need to know what it means to "triangulate" a polygon. To triangulate a polygon, pick a single vertex on the polygon and draw diagonals from that vertex to every other vertex in the polygon, forming non-overlapping triangles. For example, in this decagon, lines are drawn from the vertex in the top left of the figure to every other vertex. Part A Draw a convex quadrilateral of any type on a piece of paper, and triangulate it. How many triangles are formed?

2/28/24, 11:23 AM Lesson Activity: Investigating the Sum of Angles in Convex Polygons https://f1.app.edmentum.com/courseware-delivery/ua/125276/45610409/aHR0cHM6Ly9mMS5hcHAuZWRtZW50dW0uY29tL2x0aS92Mi9sZWFybmV… 2/8 Answer:

2/28/24, 11:23 AM Lesson Activity: Investigating the Sum of Angles in Convex Polygons https://f1.app.edmentum.com/courseware-delivery/ua/125276/45610409/aHR0cHM6Ly9mMS5hcHAuZWRtZW50dW0uY29tL2x0aS92Mi9sZWFybmV… 3/8

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2/28/24, 11:23 AM Lesson Activity: Investigating the Sum of Angles in Convex Polygons https://f1.app.edmentum.com/courseware-delivery/ua/125276/45610409/aHR0cHM6Ly9mMS5hcHAuZWRtZW50dW0uY29tL2x0aS92Mi9sZWFybmV… 4/8 Space used (includes formatting): 211 / 15000 This quadrilateral has been triangulated to form two triangles. Hide Sample Answer Part B There are in a triangle. So, how many total degrees are in a quadrilateral? Explain your reasoning using your answer from part A.

2/28/24, 11:23 AM Lesson Activity: Investigating the Sum of Angles in Convex Polygons https://f1.app.edmentum.com/courseware-delivery/ua/125276/45610409/aHR0cHM6Ly9mMS5hcHAuZWRtZW50dW0uY29tL2x0aS92Mi9sZWFybmV… 5/8 Answer: 360, This is the same for all types of quadrilaterals. Space used (includes formatting): 61 / 15000 Since two triangles are formed from triangulating a quadrilateral, there are , or , in a quadrilateral. Hide Sample Answer Part C Is it possible to triangulate a quadrilateral and get more or fewer than two triangles? Determine whether this investigation verifies that the sum of the interior angles of a quadrilateral is always . Answer: It's not possible to triangulate a quadrilateral and get more or fewer than two triangles

2/28/24, 11:23 AM Lesson Activity: Investigating the Sum of Angles in Convex Polygons https://f1.app.edmentum.com/courseware-delivery/ua/125276/45610409/aHR0cHM6Ly9mMS5hcHAuZWRtZW50dW0uY29tL2x0aS92Mi9sZWFybmV… 6/8 Space used (includes formatting): 152 / 15000 It’s not possible. No matter what type of quadrilateral it is, there will only be two triangles formed, verifying that the sum of the interior angles of a quadrilateral is always . Hide Sample Answer Part D Draw a convex pentagon, hexagon, heptagon, and octagon on a piece of paper. Triangulate each figure. Use your findings and your answers from parts A to C to complete the first five rows of the table. Then, observe the pattern to fill out the sixth row for an n -gon. Answer: Polygon Number of Sides Number of Angles Number of Triangles Total Degrees in One Triangle Total Degrees in the Polygon quadrilateral x pentagon x hexagon x x x heptagon octagon x ⋮ n -gon Space used (includes formatting): 2262 / 15000 Polygon Number of Sides Number of Angles Number of Triangles Total Degrees in One Triangle Total Degrees in the Polygon quadrilateral 4 4 2 180° 360° pentagon 5 5 3 180° 540° hexagon 6 6 4 180° 720° Hide Sample Answer

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2/28/24, 11:23 AM Lesson Activity: Investigating the Sum of Angles in Convex Polygons https://f1.app.edmentum.com/courseware-delivery/ua/125276/45610409/aHR0cHM6Ly9mMS5hcHAuZWRtZW50dW0uY29tL2x0aS92Mi9sZWFybmV… 7/8 heptagon 7 7 5 180° 900° octagon 8 8 6 180° 1080° ⋮ n -gon ⋮ n ⋮ n ⋮ n − 2 ⋮ 180° ⋮ ( n − 2) • 180° Part E Summarize your findings by stating a rule for finding the total degrees in a polygon with n sides. Answer: The sum of the interior angles in a regular polygon is given by the formula Space used (includes formatting): 88 / 15000 Hide Sample Answer Self-Evaluation How did you do? Rate your work on a scale of 1 to 5, with 5 as the highest score. Then write a brief evaluation of your work below. Note what you learned and what challenged you.

2/28/24, 11:23 AM Lesson Activity: Investigating the Sum of Angles in Convex Polygons https://f1.app.edmentum.com/courseware-delivery/ua/125276/45610409/aHR0cHM6Ly9mMS5hcHAuZWRtZW50dW0uY29tL2x0aS92Mi9sZWFybmV… 8/8 Answer: 5, I learned a lot about angles. Space used (includes formatting): 39 / 15000

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