geo 3.5 b_poly_angles_notes
TRANSCRIPT
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3.5- Part 2 Angle Measures in
Polygons
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Investigation: Sum of the interior angles
1. Draw examples of 4-sided, 5-sided, and 6-sided polygons. In each polygon, draw all the diagonals from 1 vertex.
2. Complete the table on the next slide. What is the pattern in the sum of the measures of the interior angles in any convex n-gon?
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Polygon # of Sides # of triangles
Sum of angles
Triangle 3 1 180°
Quad
Pentagon
Hexagon
n-gon n
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Polygon # of Sides # of triangles
Sum of angles
Triangle 3 1 180°
Quad 4 2 2·180=360°
Pentagon 5 3 3·180=540°
Hexagon 6 4 4·180=720°
Dodecagon 12 10 10·180= 1800
n-gon n n-2 (n-2)180°
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Polygon Interior Angles Theorem
The sum of the interior angles of a convex n-gon is
(n-2)•180°.One angle in a regular n-gon:
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n − 2( ) •180nBM #34-35
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Exterior Angles
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Polygon Exterior Angle Theorem
The sum of the measures of the exterior angles of a
convex n-gon is 360°.
BM #36
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n • interior +exterior =180n°
interior = n − 2( )180° or 180n − 360°
So, 180n − 360 + exterior = 180n−360 + exterior = 0exterior = 360°
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Example 1:
• A heptagon has 4 interior angles that measure 160° each and two interior angles that are right angles. What is the measure of the other interior angle?
BM #34
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Ex. 1 Solution:
• (n-2)180=interior sum
• (7-2)180=5•180=900°
• 4•160+2•90=640+180=820°
• 900-820=80°
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Example 2:
• Find the measure of each angle in a regular 11-gon.
BM #35
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Ex. 2 solution:
• (n-2)180=(11-2)180
• 9•180=1620°
• 1620÷11=147.3°
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Example 3:
• The measure of each exterior angle of a regular polygon is 40°. How many sides does the polygon have?
BM #36
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Ex. 3 Solution:
• 360÷40=9
• 9 sides
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Assignment:
#22 Polygon Worksheet
#23 3.5 WS (p. 301)