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Review Interior Angle Sum
New: Exterior Angle Sum
QUIZ: Prove that the diagonal connecting the vertex angles of a
kite cut the kite into two congruent triangles.
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Interior Angle Sum
Formula:
Some Problems (solve for x):
37x
141˚
3x + 90
30x - 20
153˚
116˚
x˚
4x + 101
27x + 50
20x + 15
103˚
147˚
172˚
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Working Backwards
If the sum of the angles in a polygon is 1620˚, how many sides does it have?
If the sum of the angles in a polygon is 1260˚, how many sides does it have?
If the sum of the angles in a polygon is 540˚, how many sides does it have?
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Each Interior Angle in a Regular Polygon
Formula:
Find each interior angle in each polygon:
Regular Hexagon:
Regular Decagon:
Regular Octagon:
Regular Quadrilateral:
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Working Backwards
If each interior angle in a regular polygon is 150˚, how many sides does it have?
If each interior angle in a regular polygon is 144˚, how many sides does it have?
If each interior angle in a regular polygon is 135˚, how many sides does it have?
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Exterior Angles
Definition/Diagram:
Property Compared to Interior Angles:
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Triangle:
Calculate the sum of the exterior angle measures of a triangle by
completing each step.
Step 1: Draw a triangle and extend each side to locate an exterior angle
at each vertex.
Step 2: Use the formula for the sum of interior angle measures of a
polygon and the Linear Pair Postulate to calculate the sum of the exterior
angle measures of a triangle.
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Quadrilateral
2. Calculate the sum of the exterior angle measures of a quadrilateral by
completing each step.
Step 1: Draw a quadrilateral and extend each side to locate an exterior angle
at each vertex.
Step 2: Use the formula for the sum of interior angle measures of a polygon
and the Linear Pair Postulate to calculate the sum of the exterior angle
measures of a quadrilateral.
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Pentagon
Calculate the sum of the exterior angle measures of a pentagon by
completing each step.
Step 1: Draw a pentagon and extend each side to locate an exterior
angle at each vertex.
Step 2: Use the formula for the sum of interior angle measures of a
polygon and the Linear Pair Postulate to calculate the sum of the exterior
angle measures of a pentagon.
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Hexagon
Calculate the sum of the exterior angle measures of a hexagon by
completing each step.
Step 1: Draw a hexagon and extend each side to locate an exterior angle
at each vertex.
Step 2: Use the formula for the sum of interior angle measures of a
polygon and the Linear Pair Postulate to calculate the sum of the exterior
angle measures of a hexagon.
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Putting it Together
5. Complete the table.
Number of sides of
the polygon
3
4
5
6
7
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Number of linear pairs
formed
Sum of measures of
linear pairs
Sum of measures of
interior angles
Sum of measures of
exterior angles
6. When you calculated the sum of the exterior angle measures in the 15-sided
polygon, did you need to know anything about the number of linear pairs, the
sum of the linear pair measures, or the sum of the interior angle measures of
the 15-sided polygon? Explain.
7. If a polygon has 100 sides, calculate the sum of the exterior angle measures.
Explain how you calculated your answer.
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7. If a polygon has 100 sides, calculate the sum of the exterior angle measures.
Explain how you calculated your answer.
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8. What is the sum of the exterior angle measures of an n-sided polygon?
9. If the sum of the exterior angle measures of a polygon is 360!, how many sides
does the polygon have? Explain how you got this answer.
Lesson 8.5
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Exterior and Interior Angle Measurement Interactions
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OBLEM 2
Regular Polygons
Regular Polygons
1. Calculate the measure of each exterior angle of an equilateral triangle.
Explain your reasoning.
2. Calculate the measure of each exterior angle of a square. Explain your
reasoning.
4. Calculate the measure of each exterior angle of a regular hexagon.
Explain your reasoning.
5. Complete the table shown to look for a pattern.
Number of sides of
a regular polygon
3
4
5
6
7
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2010 Carnegie Learning, Inc.
3. Calculate the measure of each exterior angle of a regular pentagon.
Explain your reasoning.
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5. Complete the table shown to look for a pattern.
Number of sides of
a regular polygon
3
4
5
6
7
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© 2010 Carnegie Learning,
Regular Polygons
Sum of measures
6. When you calculated the measure of each exterior angle in the 15-sided regular
of exterior angles
polygon, did you need to know anything about the measure of each interior
angle?
Explain.
Measure
of each
interior angle
Measure of each
exterior angle
7. If a regular polygon has 100 sides, calculate the measure of each exterior
angle. Explain how you calculated your answer.
Quadrilaterals
8. What is the measure of each exterior angle of an n-sided regular polygon?
9. If the measure of each exterior angle of a regular polygon is 18!, how many
sides does the polygon have? Explain how you calculated your answer.
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Name _____________________________________________
Date ____________________
Home Work
Exterior and Interior Angle Measurement Interactions
Sum of the Exterior Angle Measures of a Polygon
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Use the figure below to answer each question.
2
4
1
5
3
6
1. What is the sum of the measures of angles 1 and 4? Explain your reasoning.
© 2010 Carnegie Learning, Inc.
2. What is the sum of the measures of angles 2 and 5? Explain your reasoning.
3. What is the sum of the measures of angles 3 and 6? Explain your reasoning.
4. What is the sum of the measures of angles 1, 2, 3, 4, 5, and 6?
Explain your reasoning.
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Home Work
5. What is the sum of the measures of angles 1, 2, and 3? Explain your reasoning.
6. What is the difference of the sum of the measures of angles 1, 2, 3, 4, 5, and 6 and
the sum of the measures of angles 1, 2, and 3? What does this demonstrate?
7. If a regular polygon has 30 sides, what is the measure of each exterior angle?
Explain your reasoning.
, Inc.
8. The degree measure of each exterior angle of a regular octagon is represented by
the expression 7x ! 4. Solve for x.
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