Lesson 6: Polygons and Angles


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1 Lesson 6: Polygons and Angles Selected Content Standards Benchmark Assessed: G.4 Using inductive reasoning to predict, discover, and apply geometric properties and relationships (e.g., patty paper constructions, sum of the angles in a polygon) Lesson Focus In this lesson, students will discover relationships between polygons and angles. It includes the following aspects: Determine the sum of the measures of the interior angles of a triangle Recognize the relationship between interior and exterior angles Relate every convex polygon to the triangle Develop a formula to determine the sum of the interior angles of a polygon GEE 21 Connection The skills that will be addressed in this lesson include the following: Understand the relationship between triangles and all polygons Demonstrate the ability to use the interior angle theorem to determine angle measure Identify interior and exterior angles of a polygon Translating content Standards into Instruction Whether done inductively or deductively, it is important that all students understand how to find the measure of an interior angle in any polygon. This lesson is intended to help students understand the foundation of the interior and exterior angle theorem. A. The first thing students should understand is why there are 180º in a triangle. 1. Have each student draw a triangle. Do not be specific; the more types, the better. Label the angles inside the triangle with numbers. 2. On a separate sheet of paper have them draw a segment with a point near the center. They can measure the segment with a protractor to determine its measure to be 180º. 3. Have students tear off all three angles of the triangle and rearrange them with each vertex on the point on the line segment. 4. Discuss with the students to determine that everyone made the same discovery. B. Next we will relate the triangle to every convex polygon. 36
2 1. Have students draw convex polygons with 4, 5, 6, 7, 8, 9, and 10 sides. You can provide this to save a little time, but it is better to have all different sizes and shapes. 2. Have each student choose one vertex to highlight in each polygon. 3. Have students use a straight edge to connect the chosen vertex to every other vertex in the polygon. This is a good time to define diagonals. 4. Help students see that each polygon is now full of triangles. C. Begin developing a formula. 1. Students need to make a chart of the results. Name of Figure #Sides # Triangles Interior Angle Sum Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Students will see the pattern emerging from the chart. Be sure to ask them why there are 2 less triangles than sides. 3. Develop the formula (n2)180 from the discovery lesson. 4. Now, using the interior angle theorem, challenge students to find the unknown angle measures in Student Worksheet 1. D. Students should use their original diagrams to discover the exterior angle theorem. 1. Using a color different from the one use in the original diagram, have students extend one side of each angle. 2. Now have students use the supplement theorem to determine the measure of each exterior angle. 3. Extend the chart previously used to include sum of exterior angles. Name of Figure #Sides # Triangles Interior Angle Sum Exterior Angle Sum Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon
3 Sources of Evidence about Student Learning A. Have students work through the discovery lesson. B. Have students complete the student worksheet. C. Have students use a protractor to measure angles to verify their results using the angle sum theorems. GEE 21 Connection On the GEE 21 test, students may be required to A. Determine interior or exterior angle measure given partial information B. Identify formulas related to interior and exterior sums Attributes of Student Work at the GotIt level A. Students can successfully complete the student worksheet. B. Students can explain why (n2) 180 = sum of interior angles. C. Students can identify the measure of interior and exterior angles in polygons. 38
4 Lesson 6: Relationships of Polygons and Angles Student Worksheet 39
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