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Exploring Angle Relationships in Geometry

Understand the relationships of angles in triangles and polygons. Discover patterns, prove conjectures, and explore methods to calculate angles in different geometric shapes. Homework assignments included.

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Exploring Angle Relationships in Geometry

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  1. Angle Relationships Math Alliance November 30, 2010

  2. Essential Understandings • Some sequences of geometric objects change in predictable ways. • Some angles have special relationships based on their position or measure. • Polygons can be described uniquely by their sides and angles.

  3. Angle Relationships in Triangles • Draw a triangle on a piece of paper. • Find the measure of each of the 3 angles in your triangle. • Compare your angle measures with those of your table group. What patterns or relationships do you observe?

  4. Proving Your Triangle Conjecture • Other than by measuring, how could we prove that the measures of the angles in a triangle add up to 180o?

  5. Method #1: Cut off Corners • Draw a triangle on a notecard. • Cut out the triangle. • Label each vertex of the triangle (close to the vertex) • Cut off the vertices of the triangle. • Put them together. • What do you get?

  6. Method #2

  7. Extend the base • Use the Parallel Postulate (p. 604) • Look at figure 10.22 on p. 605 • What do we know? • What does this tell us? A l 4 2 5 Line l is parallel to line m 1 3 m B C

  8. What About Exterior Angles? • Draw a triangle on a piece of paper. • What is the sum of the angle measures of the 3 exterior angles in your triangle? • Compare your results with those of your table group, and make a conjecture. • How could you prove your conjecture? 1 2

  9. What About the Outside? • On a post-it, write a conjecture for what you think the rule might be for finding the measure of the exterior angles of any polygon. • How might you prove your conjecture?

  10. Sums of Exterior Angles Conjecture: Proofs • Method 1: Cut out exterior angles • Method 2: Walk-and-Turn • Method 3: Use the relation with the sum of the interior angles

  11. Other Polygons • Quadrilaterals • In your groups, draw a quadrilateral. • Measure each angle. • Find the sum of the angle measures. • Record your result. • Repeat with a polygon with more than 4 sides.

  12. What pattern(s) do you notice? • What conjecture can you make? • How could you prove your conjecture?

  13. Summary • There are 3 approaches for determining the measure of the interior angles of a polygon: • Find the triangles with a common vertex at one vertex of the polygon • Find the triangles with a common vertex in the center of the polygon • Walk and Turn

  14. Homework • Section 10.3: #’s 15, 16, 22, 41, 42, 43, 47, 49 • Turn in #’s 15, 16, 42, & 47

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