Download Lesson 6-1 - Math Slide Show

Lesson 6-1
Objective – To classify polygons and to find
relationships between the number of sides and the
measures of interior and exterior angles.
Polygon - A closed figure made up of line segments.
Circle the polygons below.
Regular Polygon - Polygon with all sides congruent
and all angles congruent.
Polygons
# of sides Name Drawing
Polygons
# of sides Name Drawing
8
Octagon
9
Nonagon
10
D
Decagon
12
Dodecagon
20
Icosagon
Regular Polygons
Drawing
Name
Regular
Octagon
Regular
Nonagon
Regular
Decagon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
Regular Polygons
Drawing
Name
Acute
Equilateral
Square
Regular
g
Pentagon
Regular
Hexagon
Regular
Heptagon
Diagonals
Diagonal - Line segment that connects two
non-adjacent vertices of a polygon.
# of diagonals
from a single vertex
Total # of diagonals
from all vertices
1
2
2
2
1
2
0
3
3
0
2
3
1
0
5
9
0
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
1
Lesson 6-1
Convex vs. Concave
(non-convex)
Convex - A polygon is convex if a segment
connecting any two points within
the figure is also fully contained
within the figure.
cave
Convex
Non-convex
Find the measure of each interior angle for the
regular polygons below.
1) Regular Pentagon
2) Regular Icosagon
5 sides
20 sides
sum of (n  2)180
angles =
(5  2)180
3 180
180   540
sum of (n  2)180
angles =
(20  2)180
18 180
180   3240
540
sum
each
angle= # sides  5
 108
3240
sum
each
angle= # sides  20
 162
Formula for Each Angle of Regular Polygon
Interior Angle =
Sum of the Interior Angles
4 sides
5 sides
6 sides
3 sides
2 180 
360
180
3 180 
540
4 180 
720
Polygon Angle Sum Theorem
The sum of the interior angle measures of a convex
polygon with n sides is (n  2)180.
1) Heptagon
3) Dodecagon
2) Nonagon
(n  2)180
(n  2)180
(n  2)180
(7  2)180
(12  2)180
(9  2)180
5 180   900 7 180   1260 10 180   1800
Sum of the Exterior Angles
120
60
60 60 120
120
Sum  3(120)
50
130 50 130
130 50 130
50
72
72
108
72
72
72
Sum  2(130)  2(50)
 360
Sum  5(72)
 360
 360
Polygon Exterior Angle Sum Theorem
The sum of the exterior angle measures (one at each
vertex) of a convex polygon with n sides is 360.
(n  2)180
n
Find the exterior angle measure for each regular
polygon below.
1) Regular
Decagon
Sum  360 Each angle  360  360  36
n
10
2) Regular
D d
Dodecagon
Sum  360 Each angle  360  360  30
n
12
3) Regular
Octagon

Sum  360 Each angle  360  360  45
n
8
Finding Interior Angles of a Regular Polygon
Find the measure of the interior angle for the
regular polygon below in two ways.
1) Regular Octadecagon (18 sides)
Interior (n  2)180
Angle =
n
(18  2)180
18
16 180 
18
Interior
Angle = 160
Exterior
Angle Sum = 360
360
Each ext. 360
 20
angle 
18
Int. 160
20
Ext.
Interior
Angle = 160
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
2
Lesson 6-1
Find the following.
1) The sum of the interior angles of a 13 sided
polygon.
sum  (n  2)180
sum  (13  2)180  11180   1980
2) The interior angle of a regular pentadecagon.
(15 sided)
Interior (n  2)180 (15  2)180 13 180 


Angle =
15
n
15
 156
Find the following.
3) The number of sides for a regular polygon with
an interior angle of 170.
Interior (n  2)180
Angle =
n

 n 170  (n  2)180
n
n
170n  180n  360
180n  180n
10n  360
 10  10
n  36 sides
Interior  170 Exterior  10
Exterior 360
Angle = n
 n
 n 10  360
 
n
10n  360
10 10
n  36 sides
Find the number of sides for each regular polygon
with the given angle.
1) Interior
2) Interior
Angle = 120
Angle = 156
Exterior  60
 n
 n  60  360
 
n
Exterior  24
 n
 n 24  360
 
n
60n  360
60
60
24n  360
24
24
n  6 sides
n  15 sides
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
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