Geometry

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Geometry

• 3.5 Angles of a Polygon

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Polygons (many angles)

• have vertices, sides, angles, and exterior
  angles
• are named by listing consecutive vertices in
  order

A
B
C
F
Hexagon ABCDEF
D
E
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Polygons

• formed by line segments, no curves
• the segments enclose space
• each segment intersects two other segments

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Polygons
Not Polygons
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Diagonal of a Polygon

• A segment connecting two nonconsecutive vertices

Diagonals
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Convex Polygons
No side collapses in toward the center
Easy test RUBBER BAND stretched around the
figure would have the same shape.
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Convex Polygons
Nonconvex Polygons
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From now on.

• When the textbook refers to polygons, it means
  convex polygons

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Polygons are classified by number of sides

• Number of sides Name of Polygon
• 3 triangle
• 4 quadrilateral
• 5 pentagon
• 6 hexagon
• 8 octagon
• 10 decagon
• n n-gon

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Interior Angles of a Polygon

• To find the sum of angle measures, divide the
  polygon into triangles
• Draw diagonals from just one vertex

4 sides, 2 triangles Angle sum 2 (180)
5 sides, 3 triangles Angle sum 3 (180)
6 sides, 4 triangles Angle sum 4 (180)
DO YOU SEE A PATTERN ?
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Interior Angles of a Polygon
4 sides, 2 triangles Angle sum 2 (180)
5 sides, 3 triangles Angle sum 3 (180)
6 sides, 4 triangles Angle sum 4 (180)
The pattern is ANGLE SUM (Number of sides
2) (180)
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Theorem

• The sum of the measures of the interior angles of
  a convex polygon with n sides is (n-2)180.

5 sides. 3 triangles. Sum of angle measures is
(5-2)(180) 3(180) 540
Example
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Exterior Angles of a Polygon
3
2
2
1
4
3
5
1
4
5
Put them together The sum 360 Works with every
polygon!
Draw the exterior angles
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Theorem

• The sum of the measures of the exterior angles of
  any convex polygon, one angle at each vertex, is
  360.

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If a polygon is both equilateral and equiangular
it is called a regular polygon
Regular Polygons
120
120
120
120
120
120
120
120
120
120
120
120
Equilateral
Equiangular
Regular
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Example 1

• A polygon has 8 sides (octagon.) Find
• The interior angle sum
• The exterior angle sum

n8, so (8-2)180 6(180)
1080 360
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Example 2
Find the measure of each interior and exterior
angle of a regular pentagon Interior (5-2)180
3(180) 540 540 108 each
5 Exterior 360 72 each 5
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Example 3

• How many sides does a regular polygon have if
• the measure of each exterior angle is 45
• 360 45 360 45n
• n n 8 8 sides an octagon
• the measure of each interior angle is 150
• (n-2)180 150 (n-2)180 150n
• n 180n 360 150n
• - 360 - 30n
• n 12 12 sides

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In summary

• Sum of interior angles
• (n-2)180
• Sum of ext. angles
• 360
• One ext. angle
• 360/n
• One int. angle
• (n 2)180/n OR supp. to 360/n
• of sides given an ext. angle
• 360/measue of ext. angle
• of sides given an int. angle
• find the ext angle(supp to int. angle)
• 360/measure of ext. angle

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Homework

• pg. 104 1-17, skip 7, bring compass