Why Study Geometry?Formal GeometryGeometric ObjectsConclusionWhy Study Geometry? Formal Geometry Geometric Objects ConclusionMATH 113Section 8.1: Basic GeometryProf. Jonathan DuncanWalla Walla UniversityWinter Quarter, 2008Why Study Geometry? Formal Geometry Geometric Objects ConclusionOutline1Why Study Geometry?2Formal Geometry3Geometric Objects4ConclusionWhy Study Geometry? Formal Geometry Geometric Objects ConclusionGeometry in HistoryOur modern concept of geometry started more than 2000 yearsago with the Greeks.Plato’s AcademyTo the Greeks, what we would callmathematics was merely a tool to thestudy of Geometry. Tradition holdsthat the inscription above the door ofPlato’s Academy read:“Let no one ignorant of Geometry enter.”Geometry is one of the fields of mathematics which is most directlyrelated to the world around us. For that reason, it is a very importantpart of elementary school mathematics.Why Study Geometry? Formal Geometry Geometric Objects ConclusionThe Study of ShapesOne way to look at geometry is as the study of shapes, theirrelationships to each other, and their properties.How is Geometry Useful?measuring land for mapsbuilding plansschematics for drawingsartistic portrayalsothers?Why Study Geometry? Formal Geometry Geometric Objects ConclusionTetris and Mathematical ThinkingGeometry is also useful in stimulating mathematics thinking. Takefor example the game of Tetris.Mathematical Thinking and TetrisDefining TermsWhat is a tetronimo? It is more than just “four squares put together.”Spatial Sense and ProbabilityWhat are good strategies for playing Tetris?StatisticsMeasure improvement by recording scores and comparing early scores to later scores.CongruenceWhich pieces are the same and which are actually different?Problem SolvingHow many Tetris pieces are there?TessellationWhich tetronimo will cover a surface with no gaps?Geometry and AlgebraHow does the computer version of Tetris work?Why Study Geometry? Formal Geometry Geometric Objects ConclusionTwo Types of GeometryTraditionally, geometry in education can be divided into twodistinct types.Types of GeometryFormal GeometryThis type of geometry is similar to that studied in ancientGreece in which everything is proven from a set of basicaxioms.Informal or Conceptual GeometryIn this type of geometry focus is placed on shapes andrelationships and not on formal axioms and proofs.We will spend a little time with Formal Geometry before going onto talk more about conceptual geometry.Why Study Geometry? Formal Geometry Geometric Objects ConclusionEuclid’s PostulatesWe can not prove everything! We must have some starting pointto any formal system.Euclid’s PostulatesOne of the most famous bo oks in history is Euclid’s Elements. In it theGreek mathematician Euclid presented his five postulates for geometry.Postulates are statements which are to be accepted as true without proof.1A straight line may be drawn between any two points.2A piece of a straight line may be extended indefinitely.3A c ircle may be drawn with any given radius and an arbitrary center.4All right angles are equal.5If a straight line crossing two straight lines makes the interior angleson the same side less than two right angles, the two straight lines, ifextended indefinitely, meet on that side on which the angles lessthan two right angles lie.Why Study Geometry? Formal Geometry Geometric Objects ConclusionEuclid’s Fifth PostulateTo understand the consequences of stating postulates, considerEuclid’s fifth postulate. This is often called the parallel postulateand has been controversial.Alternatives to the Parallel PostulateThe following are alternatives to the parallel postulate which states that two lines which are not parallel must intersect.There exists a pair of similar non-congruent triangles.There exists a pair of straight lines everywhere equidistant from one another.There exists a circle through any three non-colinear points.If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.If a straight line intersects one of two parallel lines it will intersect the other.Straight lines parallel to a third line are parallel to each other.Two straight lines that intersect one another cannot be parallel to a third line.There is no upper limit to the area of a triangle.Why Study Geometry? Formal Geometry Geometric Objects ConclusionBasic Objects in GeometryWe now turn to the more conceptual questions in geometry. Thoseinvolving basic objects and their relationships.Basic Geometric ObjectsThe following objects in geometry can not be formally defined, butwe must agree on what the terms mean.Pointspoints have no dimensions but they do have a locationLineslines are straight, extend infinitely in two directions, and canbe thought of as being made up of points.Planea plane is a flat surface which extends infinitely in twodimensions.Why Study Geometry? Formal Geometry Geometric Objects ConclusionColinearityExample1How many lines are there through a single point?2How many lines are there through two distinct points?3How many lines are there through three distinct points?Colinear PointsA set of points is colinear if there is a single line through all of thepoints. (Note: Every set of two points is colinear.)ExampleDraw a set of three points which are colinear and another set ofthree points which are not colinear.Why Study Geometry? Formal Geometry Geometric Objects ConclusionCoplanarityQuestions line those we asked about lines can be asked aboutplanes as well.Example1How many planes are there through a single point?2How many planes are there through two points?3How many planes are there through three points?Coplanar PointsA set of points is said to be coplanar if there is a plane containingall points in the set.ExampleCan you find a set of points which are not coplanar?Why Study Geometry? Formal Geometry Geometric Objects ConclusionFrom a Line To. . .Using the basic object of a line, we can define several new objects.Line SegmentA line segment is a subset of the line which contains two pointson the line, called endpoints, and all parts of the line betweenthese two points.RaysA ray is a subset of a line that contains a specific point, called theendpoint, and all points on the line on one side of the endpoint.ExampleDraw an example of a line, line segment, and a ray and name eachobject using point names.Why Study Geometry? Formal Geometry Geometric Objects ConclusionRelationships Between LinesTwo lines can