Math 310 - PowerPoint PPT Presentation

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Math 310

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Title: Math 310


1
Math 310
  • 9.2
  • Polygons

2
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 cant be
    split into two disjoint sets.

3
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.

4
Closed Curve
  • Def
  • A closed curve is a curve that has the same
    beginning and ending point.

5
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.

6
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.

7
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.

8
Types of Polygons
  • Polygons are classified, primarily, according to
    the number of sides that they have. This is
    summarized in the following table.

9
Types of Polygons (cont)
Polygon Number of Sides (or vertices)
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
n-gon n
10
Parts of a Polygon
  • Side
  • Vertex
  • Interior angle
  • Exterior angle
  • diagonal

11
Interior Angle
  • Def
  • An interior angle of a polygon is created by
    three consecutive vertices.

12
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.

13
Diagonal
  • Def
  • A diagonal of a polygon is a segment connecting
    any two non-consecutive vertices.

14
Ex
exterior angle
exterior angle
15
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

16
Ex.
17
Regular Polygon
  • Def
  • A polygon is called regular if all its angles and
    sides are congruent.

18
Ex
19
Further Classification of Polygons
  • Triangles
  • Quadrilaterals

20
Triangles
  • Right triangle
  • Acute triangle
  • Obtuse triangle
  • Scalene triangle
  • Isosceles triangle
  • Equilateral triangle

21
Right Triangle
  • Def
  • A right triangle contains exactly one right angle.

22
Acute Obtuse Triangle
  • Def
  • An acute triangle contains all acute angles while
    an obtuse triangle contains exactly one obtuse
    angle.

23
Scalene Triangle
  • Def
  • A scalene triangle has three non-congruent sides.
    (ie all the sides are of different length.)

24
Isosceles Triangle
  • Def
  • An isosceles triangle has at least two congruent
    sides.

25
Equilateral Triangle
  • Def
  • An equilateral triangle is a triangle with all
    sides congruent.

26
Quadrilaterals
  • Trapezoid
  • Kite
  • Isosceles trapezoid
  • Parallelogram
  • Rectangle
  • Rhombus
  • Square

27
Trapezoid
  • Def
  • A trapezoid is a quadrilateral with at least one
    pair of parallel sides.

28
Isosceles Trapezoid
  • Def
  • An isosceles trapezoid is a trapezoid with the
    non-parallel sides being congruent

29
Kite
  • Def
  • A quadrilateral with two non-overlapping pairs of
    congruent adjacent sides.

NON-VERTEX ANGLES
VERTEX ANGLES
30
Parallelogram
  • Def
  • A parallelogram is a quadrilateral with opposite
    sides parallel.

31
Rectangle
  • Def
  • A rectangle is a parallelogram with a right
    angle.

32
Rhombus
  • A rhombus is a quadrilateral with four congruent
    sides.

33
Square
  • A square is a rhombus with a right angle.

34
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.

35
Quadrilateral Hierarchy
Quadrilateral
Trapezoid
Kite
Isosceles Trapezoid
Parallelogram
Rhombus
Rectangle
Square
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