Angle Relationships

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?

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Method 2
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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
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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

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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.

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Sides Total Angle Measure Avg. Angle Measure
3 180 60
4
5
6
7
8



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

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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

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Homework

• Section 10.3 s 15, 16, 22, 41, 42, 43, 47, 49
  
• Turn in s 15, 16, 42, 47