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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:
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.
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.
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.