Download Geometry 6-1 Angles of Polygons

Geometry 6-1 Angles of Polygons
A diagonal
of a polygon is a segment that connects
any two nonconsecutive
vertices.
F
G
A
B
F
E
B
E
GC is a diagonal.
D
Find the sum of the measures of the interior angles of a
convex decagon.
decagon: n = 10
sides
Find the measure of each interior angle in this pentagon.
2x + 142 + 2x + 3x + 14 + 3x + 14 = (5 - 2) 180
J
142
H
10x + 170 = 540
10x = 370
x = 37
2x
2x K
H = 2(37) = 74
K = 2(37) = 74
L = 3(37) + 14 = 105
M = 3(37) + 14 = 105
Hendecagon = 11
sides
1620 / 11 = 147.273
We can go the other way as well. If we know the measure of
the interior angle, we can find the number of sides. We just
need to reverse the process.
J = 142
The measure of an interior angle of a regular polygon is 135.
Find the number of sides in the polygon.
135x = (x - 2) 180
135x = 180x - 360
-45x = -360
x=8
Find the measure of each interior angle of a regular hendecagon.
(11 - 2) 180 = 9 180 = 1620
Sum = (10 - 2) 180 = 8 180 = 1440
3x + 14
If we draw in all of the diagonals from
a given vertex, then the polygon is
divided into triangles. We can use
this idea to find the sum of the
interior angles of any polygon.
D
C
Theorem 6.1 - Polygon Interior Angles Sum: The sum of
the interior angle measures of an n-sided convex polygon is
(n - 2) 180
.
C
M 3x + 14 L
G
A
Theorem 6.2 - Polygon Exterior Angles Sum: The sum of the
measures of the exterior angles of a convex polygon is 360
.
6x
Find the value of x in the diagram.
2x + 9x + 6x + 139 = 360
17x = 221
x = 13
139
2x
9x
Find the measure of each exterior angle of a regular dodecagon.
dodecagon = 12
sides
12x = 360
x = 30