Download Math 2 Geometry Based on Elementary Geometry, 3rd ed, by

Math 2 Geometry
Based on Elementary Geometry, 3rd ed, by Alexander & Koeberlein
2.5
Convex Polygons
Definition
A polygon is a closed plane figure whose
sides are line segments that intersect only
at the endpoints.
In this text only: convex polygons
(interior angles measure from 0 to 180)
Examples?
Polygon Examples
Convex
Polygons
Concave Polygons
Not Polygons
Classify by number of sides
Triangle
(3)
Quadrilateral (4)
Pentagon
(5)
Classify by number of sides
Hexagon (6)
Heptagon (7)
Octagon (8)
Classify by number of sides
Nonagon (9)
Decagon (10)
Informal Definition
A diagonal of a polygon is a line segment
that joins two nonconsecutive vertices.
B
A
C
E
D
Theorem 2.5.1
The total number of diagonals D in a
polygon of n sides is given by the formula:
n (n  3)
D
2
Use the formula to find the number of
diagonals of:
•
a triangle
•
a quadrilateral
•
a pentagon
n (n  3)
D
2
Theorem 2.5.2
The sum S of the measures of the interior
angles of a polygon with n sides is given by:
S = (n – 2) 180
•
Informal Definitions
• Equiangular polygon: All angles are
congruent.
• Equilateral polygon: The measure of all
sides are equal.
Definition
A regular polygon is a polygon that is both
equilateral and equiangular.
Notice: A square is a regular quadrilateral.
Not Regular Polygons
Notice it is possible for Polygons to be either
equilateral or equiangular but not both.
Rhombus
Rectangle
Corollary 2.5.3
The measure I of each interior angle of a
regular polygon of n sides is:
(n  2) 180
I
n
Corollary 2.5.4
The sum of the four interior angles of a
quadrilateral is 360
Corollary 2.5.5
The sum of the measures of the exterior
angles of a polygon, one at each vertex, is
360.
Corollary 2.5.6
The measure E of each exterior angle of a
regular polygon of n sides is:
360
E
n
Informal Definition
A polygram is the star-shaped figure that
results when the sides of certain polygons
are extended.
Pentagram
Hexagram