Transcript 6.1 Polygons - Adair County Schools

3.4 The Polygon AngleSum Theorems
Geometry
Mrs. Loy
Objectives:
• To Classify polygons
• To find the sums of the measures of the
interior and exterior angles of polygons.
Q
VERTEX
R
SIDE
Definitions:
P
S
VERTEX
T
• Polygon—a plane figure that meets the following
conditions:
– It is formed by 3 or more segments called sides, such
that no two sides with a common endpoint are collinear.
– Each side intersects exactly two other sides, one at each
endpoint.
• Vertex – each endpoint of a side. Plural is
vertices. You can name a polygon by listing its
vertices consecutively. For instance, PQRST and
QPTSR are two correct names for the polygon
above.
Example 1: Identifying Polygons
• State whether the figure is
a polygon. If it is not,
explain why.
• Not D – has a side that
isn’t a segment – it’s an
arc.
• Not E– because two of the
sides intersect only one
other side.
• Not F because some of its
sides intersect more than
two sides/
A
C
B
F
E
D
Figures A, B, and C are
polygons.
Polygons are named by the number
of sides they have – MEMORIZE
Number of sides
Type of Polygon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
Polygons are named by the number
of sides they have – MEMORIZE
Number of sides
Type of Polygon
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
n
n-gon
Convex or concave?
• Convex if no line that
contains a side of the
polygon contains a point
in the interior of the
polygon.
• Concave or non-convex if
a line does contain a side
of the polygon containing
a point on the interior of
the polygon.
See how this crosses
a point on the inside?
Concave.
See how it doesn’t go on the
Inside-- convex
Convex or concave?
• Identify the polygon
and state whether it is
convex or concave.
A polygon is EQUILATERAL
If all of its sides are congruent.
A polygon is EQUIANGULAR
if all of its interior angles are
congruent.
A polygon is REGULAR if it is
equilateral and equiangular.
P
80°
Ex. : Interior Angles of a
Quadrilateral
70°
x°
Q
x°+ 2x° + 70° + 80° = 360°
3x + 150 = 360
3x = 210
x = 70
S
2x°
R
Sum of the measures of int. s of a
quadrilateral is 360°
Combine like terms
Subtract 150 from each side.
Divide each side by 3.
Find m Q and mR.
mQ = x° = 70°
mR = 2x°= 140°
►So, mQ = 70° and mR = 140°
Investigation Activity
• Sketch polygons with 4, 5,
6, 7, and 8 sides
• Divide Each Polygon into
triangles by drawing all
diagonals that are possible
from one vertex
• Multiply the number of
triangles by 180 to find
the sum of the measures of
the angles of each
polygon.
1) Look for a pattern.
Describe any that
you have found.
2) Write a rule for the
sum of the measures
of the angles of an ngon
Polygon Angle-Sum theorem
• The sum of the
measures of the angles
of an n-gon is
(n-2)180
• Ex: Find the sum of
the measures of the
angles of a 15-gon
• Sum = (n-2)180
•
= (15-2)180
•
= 13*180
= 2340
Example
• The sum of the interior
angles of a polygon is
9180. How many
sides does the polygon
have?
•
•
•
•
•
Sum = (n-2)180
9180 = (n-2)180
51 = n-2
53 = n
The polygon has 53
sides.
Polygon Exterior Angle-Sum
Theorem
• The sum of the
measures of the
exterior angles of a
polygon, one at each
vertex, is 360.
• An equilateral polygon
has all sides congruent
• An equiangular
polygon has all angles
congruent
• A regular polygon is
both equilateral and
equiangular.
Example
• The measure of an
exterior angle of a
regular polygon is 36.
Find the measure of an
interior angle, and find
the number of sides.
•
•
•
•
•
Exterior angles = 360
Since regular,
n*36 = 360
n = 10
Since exterior angle =
36, interior angle
• 180-26 = 144
Assignment
• Pg 147-150
#32, 34, 40-46 even,
47-49, 71-86 all