Download 10-1: Polygons

NOTES: Polygons (Glencoe 10-1)
The term polygon is derived from the Greek word
meaning “________-____________.”
A polygon is a ____________ figure formed by a
____________ number of ________________
segments such that


A ____________ polygon is a polygon such that no
line containing a side of the polygon contains a
point on the interior of the polygon. A polygon that
is not convex is nonconvex or ______________.
Polygons may be classified by the number of sides
they have. In general, a polygon with n sides is
called an n-gon.
Number of Sides
Polygon
3
4
5
6
7
8
9
10
12
n
When referring to a polygon, we use its name and
S
list the vertices in consecutive order.
Pentagon __________ and
R
T
pentagon __________ are two
correct names for the polygon
at the right; there are many others.
V
U
A regular polygon is a ____________ polygon
with ______ __________ __________________
and ______ ____________ _________________.
Example:
Name and classify each polygon
(a.) by the number of sides
(b.) as convex or concave
(c.) as regular or not regular
T
U
W
V
O
M
P
N
L
K
R
S
Q
Theorem 10-1: Interior Angle Sum Theorem
If a convex polygon has n sides and S is the sum of
the measures of its interior angles, then
S = __________________.
Example:
Find the sum of the measures of the interior angles
of each convex polygon.
(1.) decagon
(2.) 21-gon
Example:
Find the measure of each interior angle of a regular
hexagon.
Theorem 10-2: Exterior Angle Sum Theorem
If a polygon is convex, then the sum of the
measures of the exterior angles, one at each vertex,
is ______.
Example:
The measures of an exterior angle of a regular
polygon is given. Find the number of sides of the
polygon.
(1.) 30
(2.) 8
Example:
The number of sides of a regular polygon is given.
Find the measures of an interior angle and an
exterior angle of the polygon.
(1.) 8
(2.) a
Example:
The measure of an interior angle of a regular
polygon is given. Find the number of sides in each
polygon.
(1.) 120
(2.) 150
Example:
The measures of the interior angles of a pentagon
are x, 3x, 2x - 1, 6x - 5, and 4x + 2. Find the
measure of each angle.
NOTES: Polygons (Glencoe 10-1)
The term polygon is derived from the Greek word
meaning “many-angled.”
A polygon is a closed figure formed by a finite
number of coplanar segments such that
 the sides that have a common endpoint are
noncollinear, and
 each side intersects exactly two other sides, but
only at their endpoints.
give examples & non-examples of polygons
A convex polygon is a polygon such that no line
containing a side of the polygon contains a point on
the interior of the polygon. A polygon that is not
convex is nonconvex or concave.
Polygons may be classified by the number of sides
they have. In general, a polygon with n sides is
called an n-gon.
Number of Sides
Polygon
3
triangle
4
quadrilateral
5
pentagon
6
hexagon
7
heptagon
8
octagon
9
nonagon
10
decagon
12
dodecagon
n
n-gon
When referring to a polygon, we use its name and
S
list the vertices in consecutive order.
Pentagon RSTUV and
R
T
pentagon TSRUV are two
correct names for the polygon
at the right; there are many others.
V
U
A regular polygon is a convex polygon with all
sides congruent and all angles congruent.
Example:
Name and classify each polygon
(a.) by the number of sides
(b.) as convex or concave
(c.) as regular or not regular
T
U
W
V
O
M
P
N
L
K
R
a) quadrilateral
b) convex
c) not regular (all
angles not congruent)
S
Q
a) nonagon
b) concave
("caves in" at S
and N)
c) not regular (not
convex)
Theorem 10-1: Interior Angle Sum Theorem
If a convex polygon has n sides and S is the sum of
the measures of its interior angles, then
S = 180(n - 2).
Example:
Find the sum of the measures of the interior angles
of each convex polygon.
(1.) decagon
10 sides
S = 180(10 - 2) = 1440
(2.) 21-gon
21 sides
S = 180(21 - 2) = 3420
Example:
Find the measure of each interior angle of a regular
hexagon.
a hexagon has 6 sides
sum = 180(6 - 4) = 720
each angle = 720  6 = 120
Theorem 10-2: Exterior Angle Sum Theorem
If a polygon is convex, then the sum of the
measures of the exterior angles, one at each vertex,
is 360.
Example:
The measures of an exterior angle of a regular
polygon is given. Find the number of sides of the
polygon.
(1.) 30
(2.) 8
360  30 = 12 sides
360  8 = 45 sides
Example:
The number of sides of a regular polygon is given.
Find the measures of an interior angle and an
exterior angle of the polygon.
(1.) 8
each exterior angle = 360  8 = 45
each interior angle = 180 - 45 = 135
(2.) a
each exterior angle = 360/a
each interior angle = 180 - (360/a)
Example:
The measure of an interior angle of a regular
polygon is given. Find the number of sides in each
polygon.
(1.) 120
an exterior angle = 180 - 120 = 60
# of sides = 360  60 = 6 sides
(2.) 150
an exterior angle = 180-150 = 30
# of sides = 360  30 = 12 sides
Example:
The measures of the interior angles of a pentagon
are x, 3x, 2x - 1, 6x - 5, and 4x + 2. Find the
measure of each angle.
sum of interior angles = 180(5 - 2) = 540
x + 3x + 2x - 1 + 6x - 5 + 4x + 2 = 540
16x - 4 = 540
16x = 544
x = 34
3(34) = 102; 2(34) - 1 = 67; 6(34) - 5 = 199;
4(34) + 2 = 138