Math 310Math 310

9.29.2

PolygonsPolygons

Curve & ConnectedCurve & Connected

IdeaIdea

The idea of a The idea of a curvecurve is something you is something you could draw on paper without lifting could draw on paper without lifting your pencil.your pencil.

The idea of The idea of connectedconnected is that a set can’t is that a set can’t be split into two disjoint sets.be split into two disjoint sets.

Simple CurveSimple Curve

DefDef

A A simple curvesimple curve is a curve that does is a curve that does not cross itself, with the possible not cross itself, with the possible exception of having the same exception of having the same beginning and ending points.beginning and ending points.

Closed CurveClosed Curve

DefDef

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

PolygonPolygon

DefDef

A A polygonpolygon is a planar figure, that is a is a planar figure, that is a simple closed curve consisting simple closed curve consisting entirely of straight segments. entirely of straight segments. Where two segments meet is called a Where two segments meet is called a vertexvertex of the polygon. Each of the polygon. Each segment is called a segment is called a sideside of the of the polygon.polygon.

Convex and Concave Convex and Concave PolygonsPolygons

DefDef

A A convex polygonconvex polygon is one in which every is one in which every point in the interior of the polygon can point in the interior of the polygon can be joined by a line segment that stays be joined by a line segment that stays entirely inside the polygon. A entirely inside the polygon. A concave concave polygonpolygon would be one in which there would be one in which there exist at least two points, such that exist at least two points, such that when joined by a line segment, the when joined by a line segment, the segment passes outside of the polygon.segment passes outside of the polygon.

Polygonal RegionPolygonal Region

DefDef

A polygon divides the plane its in, into A polygon divides the plane its in, into three regions, the interior, exterior, three regions, the interior, exterior, and the curve itself. The interior and the curve itself. The interior and the curve itself comprise the and the curve itself comprise the polygonal regionpolygonal region we are concerned we are concerned with.with.

Types of PolygonsTypes of Polygons

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

Types of Polygons (cont)Types of Polygons (cont)

PolygonPolygon Number of Sides (or Number of Sides (or vertices)vertices)

TriangleTriangle 33

QuadrilateralQuadrilateral 44

PentagonPentagon 55

HexagonHexagon 66

HeptagonHeptagon 77

OctagonOctagon 88

NonagonNonagon 99

DecagonDecagon 1010

n-gonn-gon nn

Parts of a PolygonParts of a Polygon

SideSide VertexVertex Interior angleInterior angle Exterior angleExterior angle diagonaldiagonal

Interior AngleInterior Angle

DefDef

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

Exterior AngleExterior Angle

DefDef

An exterior angle of a polygon is An exterior angle of a polygon is created by extending a side out from created by extending a side out from the polygon and reading the angle the polygon and reading the angle between the contiguous side.between the contiguous side.

DiagonalDiagonal

DefDef

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

ExEx

A

E

D

C

B

vertex side

interior angle

interior

exterior

exterior angle

exterior angle

A

E

D

C

BF

G

diagonal

CongruencyCongruency

IdeaIdea

Two segments are congruent if the Two segments are congruent if the tracing of one can be fitted exactly tracing of one can be fitted exactly onto the other. Two angles are onto the other. Two angles are congruent if their measures are the congruent if their measures are the same. Two polygons are congruent if same. Two polygons are congruent if all their parts are congruent.all their parts are congruent.

Notation: Notation:

A

B

C

D

Ex.Ex.

CDAB

E

FG

H

J

I

HIJEFG

Regular PolygonRegular Polygon

DefDef

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

ExEx

Further Classification of Further Classification of PolygonsPolygons

TrianglesTriangles QuadrilateralsQuadrilaterals

TrianglesTriangles

Right triangleRight triangle Acute triangleAcute triangle Obtuse triangleObtuse triangle Scalene triangleScalene triangle Isosceles triangleIsosceles triangle Equilateral triangleEquilateral triangle

Right TriangleRight Triangle

DefDef

A right triangle contains exactly one A right triangle contains exactly one right angle.right angle.

Acute & Obtuse TriangleAcute & Obtuse Triangle

DefDef

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

Scalene TriangleScalene Triangle

DefDef

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

Isosceles TriangleIsosceles Triangle

DefDef

An isosceles triangle has at least two An isosceles triangle has at least two congruent sides.congruent sides.

Equilateral TriangleEquilateral Triangle

DefDef

An equilateral triangle is a triangle An equilateral triangle is a triangle with all sides congruent.with all sides congruent.

QuadrilateralsQuadrilaterals

TrapezoidTrapezoid KiteKite Isosceles trapezoidIsosceles trapezoid ParallelogramParallelogram RectangleRectangle RhombusRhombus Square Square

TrapezoidTrapezoid

DefDef

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

Isosceles TrapezoidIsosceles Trapezoid

DefDef

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

KiteKite

DefA quadrilateral with

two non-overlapping pairs of congruent adjacent sides.

VERTEX

ANGLES

NON-VERTEX

ANGLES

ParallelogramParallelogram

DefDef

A parallelogram is a quadrilateral with A parallelogram is a quadrilateral with opposite sides parallel.opposite sides parallel.

RectangleRectangle

DefDef

A rectangle is a parallelogram with a A rectangle is a parallelogram with a right angle. right angle.

RhombusRhombus

A rhombus is a quadrilateral with four A rhombus is a quadrilateral with four congruent sides.congruent sides.

SquareSquare

A square is a rhombus with a right A square is a rhombus with a right angle.angle.

Quadrilateral HierarchyQuadrilateral Hierarchy

By looking at the definitions, we see that By looking at the definitions, we see that some of the quadrilateral definitions some of the quadrilateral definitions fulfill one or more of the other fulfill one or more of the other definitions automatically. For example, definitions automatically. For example, the definition of a square also satisfies the definition of a square also satisfies the definition of a rhombus. This allows the definition of a rhombus. This allows us to set up a hierarchy among the us to set up a hierarchy among the quadrilaterals which will be very useful quadrilaterals which will be very useful when discussing their various when discussing their various properties.properties.

Quadrilateral Quadrilateral HierarchyHierarchy

Quadrilateral

Kite

Trapezoid

Isosceles Trapezoid

Parallelogram

Rhombus

Rectangle

Square