Download 9-2

Math 310
9.2
Polygons
Curve & Connected
Idea
The idea of a curve is something you could draw
on paper without lifting your pencil.
The idea of connected is that a set can’t be split
into two disjoint sets.
Simple Curve
Def
A simple curve is a curve that does not cross
itself, with the possible exception of having the
same beginning and ending points.
Closed Curve
Def
A closed curve is a curve that has the same
beginning and ending point.
Polygon
Def
A polygon is a planar figure, that is a simple
closed curve consisting entirely of straight
segments. Where two segments meet is called a
vertex of the polygon. Each segment is called a
side of the polygon.
Convex and Concave Polygons
Def
A convex polygon is one in which every point in
the interior of the polygon can be joined by a
line segment that stays entirely inside the
polygon. A concave polygon would be one in
which there exist at least two points, such that
when joined by a line segment, the segment
passes outside of the polygon.
Polygonal Region
Def
A polygon divides the plane its in, into three
regions, the interior, exterior, and the curve
itself. The interior and the curve itself comprise
the polygonal region we are concerned with.
Types of Polygons
Polygons are classified, primarily, according to the
number of sides that they have. This is
summarized in the following table.
Types of Polygons (cont)
Polygon
Number of Sides (or vertices)
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
n-gon
3
4
5
6
7
8
9
10
n
Parts of a Polygon





Side
Vertex
Interior angle
Exterior angle
diagonal
Interior Angle
Def
An interior angle of a polygon is created by three
consecutive vertices.
Exterior Angle
Def
An exterior angle of a polygon is created by
extending a side out from the polygon and
reading the angle between the contiguous side.
Diagonal
Def
A diagonal of a polygon is a segment connecting
any two non-consecutive vertices.
exterior
interior
A
Ex
B
interior
angle
E
F
C
vertex
D
A
exterior
angle
B
G
side
E
C
exterior
angle
diagonal
D
Congruency
Idea
Two segments are congruent if the tracing of one
can be fitted exactly onto the other. Two angles
are congruent if their measures are the same.
Two polygons are congruent if all their parts are
congruent.
Notation: 
Ex.
B
E
A
C
F
D
AB  CD
G
H
I
EFG  HIJ
J
Regular Polygon
Def
A polygon is called regular if all its angles and
sides are congruent.
Ex
Further Classification of
Polygons


Triangles
Quadrilaterals
Triangles






Right triangle
Acute triangle
Obtuse triangle
Scalene triangle
Isosceles triangle
Equilateral triangle
Right Triangle
Def
A right triangle contains exactly one right angle.
Acute & Obtuse Triangle
Def
An acute triangle contains all acute angles while an
obtuse triangle contains exactly one obtuse
angle.
Scalene Triangle
Def
A scalene triangle has three non-congruent sides.
(ie all the sides are of different length.)
Isosceles Triangle
Def
An isosceles triangle has at least two congruent
sides.
Equilateral Triangle
Def
An equilateral triangle is a triangle with all sides
congruent.
Quadrilaterals







Trapezoid
Kite
Isosceles trapezoid
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Def
A trapezoid is a quadrilateral with at least one pair
of parallel sides.
Isosceles Trapezoid
Def
An isosceles trapezoid is a trapezoid with the nonparallel sides being congruent
Kite
Def
A quadrilateral with two
non-overlapping pairs
of congruent adjacent
sides.
Parallelogram
Def
A parallelogram is a quadrilateral with opposite
sides parallel.
Rectangle
Def
A rectangle is a parallelogram with a right angle.
Rhombus
A rhombus is a quadrilateral with four congruent
sides.
Square
A square is a rhombus with a right angle.
Quadrilateral Hierarchy
By looking at the definitions, we see that some of
the quadrilateral definitions fulfill one or more
of the other definitions automatically. For
example, the definition of a square also satisfies
the definition of a rhombus. This allows us to
set up a hierarchy among the quadrilaterals
which will be very useful when discussing their
various properties.
Quadrilateral
Hierarchy
Quadrilateral
Trapezoid
Kite
Parallelogram
Isosceles
Trapezoid
Rhombus
Rectangle
Square