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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

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math 310

Math 310

9.2

Polygons

curve connected
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
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
Closed Curve

Def

A closed curve is a curve that has the same beginning and ending point.

polygon
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
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
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
Types of Polygons

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

parts of a polygon
Parts of a Polygon
  • Side
  • Vertex
  • Interior angle
  • Exterior angle
  • diagonal
interior angle
Interior Angle

Def

An interior angle of a polygon is created by three consecutive vertices.

exterior angle
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
Diagonal

Def

A diagonal of a polygon is a segment connecting any two non-consecutive vertices.

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exterior

interior

interior angle

side

vertex

diagonal

Ex

exterior angle

exterior angle

congruency
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:

regular polygon
Regular Polygon

Def

A polygon is called regular if all its angles and sides are congruent.

further classification of polygons
Further Classification of Polygons
  • Triangles
  • Quadrilaterals
triangles
Triangles
  • Right triangle
  • Acute triangle
  • Obtuse triangle
  • Scalene triangle
  • Isosceles triangle
  • Equilateral triangle
right triangle
Right Triangle

Def

A right triangle contains exactly one right angle.

acute obtuse triangle
Acute & Obtuse Triangle

Def

An acute triangle contains all acute angles while an obtuse triangle contains exactly one obtuse angle.

scalene triangle
Scalene Triangle

Def

A scalene triangle has three non-congruent sides. (ie all the sides are of different length.)

isosceles triangle
Isosceles Triangle

Def

An isosceles triangle has at least two congruent sides.

equilateral triangle
Equilateral Triangle

Def

An equilateral triangle is a triangle with all sides congruent.

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

Def

A trapezoid is a quadrilateral with at least one pair of parallel sides.

isosceles trapezoid
Isosceles Trapezoid

Def

An isosceles trapezoid is a trapezoid with the non-parallel sides being congruent

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Kite

Def

A quadrilateral with two non-overlapping pairs of congruent adjacent sides.

NON-VERTEX ANGLES

VERTEX ANGLES

parallelogram
Parallelogram

Def

A parallelogram is a quadrilateral with opposite sides parallel.

rectangle
Rectangle

Def

A rectangle is a parallelogram with a right angle.

rhombus
Rhombus

A rhombus is a quadrilateral with four congruent sides.

square
Square

A square is a rhombus with a right angle.

quadrilateral hierarchy
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 hierarchy35

Quadrilateral

Quadrilateral Hierarchy

Trapezoid

Kite

Parallelogram

Isosceles Trapezoid

Rhombus

Rectangle

Square