Why Study Geometry? Formal Geometry Geometric Objects Conclusion MATH 113 Section 8.1: Basic Geometry Prof. Jonathan Duncan Walla Walla University Winter Quarter, 2008 Why Study Geometry? Formal Geometry Geometric Objects Conclusion Outline 1 Why Study Geometry? 2 Formal Geometry 3 Geometric Objects 4 Conclusion Why Study Geometry? Formal Geometry Geometric Objects Conclusion Geometry in History Our modern concept of geometry started more than 2000 years ago with the Greeks. Plato’s Academy To the Greeks, what we would call mathematics was merely a tool to the study of Geometry. Tradition holds that the inscription above the door of Plato’s Academy read: “Let no one ignorant of Geometry enter.” Geometry is one of the fields of mathematics which is most directly related to the world around us. For that reason, it is a very important part of elementary school mathematics. Why Study Geometry? Formal Geometry Geometric Objects Conclusion The Study of Shapes One way to look at geometry is as the study of shapes, their relationships to each other, and their properties. How is Geometry Useful? measuring land for maps building plans schematics for drawings artistic portrayals others? Why Study Geometry? Formal Geometry Geometric Objects Conclusion Tetris and Mathematical Thinking Geometry is also useful in stimulating mathematics thinking. Take for example the game of Tetris. Mathematical Thinking and Tetris Defining Terms What is a tetronimo? It is more than just “four squares put together.” Spatial Sense and Probability What are good strategies for playing Tetris? Statistics Measure improvement by recording scores and comparing early scores to later scores. Congruence Which pieces are the same and which are actually different? Problem Solving How many Tetris pieces are there? Tessellation Which tetronimo will cover a surface with no gaps? Geometry and Algebra How does the computer version of Tetris work? Why Study Geometry? Formal Geometry Geometric Objects Conclusion Two Types of Geometry Traditionally, geometry in education can be divided into two distinct types. Types of Geometry Formal Geometry This type of geometry is similar to that studied in ancient Greece in which everything is proven from a set of basic axioms. Informal or Conceptual Geometry In this type of geometry focus is placed on shapes and relationships and not on formal axioms and proofs. We will spend a little time with Formal Geometry before going on to talk more about conceptual geometry. Why Study Geometry? Formal Geometry Geometric Objects Conclusion Euclid’s Postulates We can not prove everything! We must have some starting point to any formal system. Euclid’s Postulates One of the most famous books in history is Euclid’s Elements. In it the Greek mathematician Euclid presented his five postulates for geometry. Postulates are statements which are to be accepted as true without proof. 1 A straight line may be drawn between any two points. 2 A piece of a straight line may be extended indefinitely. 3 A circle may be drawn with any given radius and an arbitrary center. 4 All right angles are equal. 5 If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles less than two right angles lie. Why Study Geometry? Formal Geometry Geometric Objects Conclusion Euclid’s Fifth Postulate To understand the consequences of stating postulates, consider Euclid’s fifth postulate. This is often called the parallel postulate and has been controversial. Alternatives to the Parallel Postulate The 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 Conclusion Basic Objects in Geometry We now turn to the more conceptual questions in geometry. Those involving basic objects and their relationships. Basic Geometric Objects The following objects in geometry can not be formally defined, but we must agree on what the terms mean. Points points have no dimensions but they do have a location Lines lines are straight, extend infinitely in two directions, and can be thought of as being made up of points. Plane a plane is a flat surface which extends infinitely in two dimensions. Why Study Geometry? Formal Geometry Geometric Objects Conclusion Colinearity Example 1 How many lines are there through a single point? 2 How many lines are there through two distinct points? 3 How many lines are there through three distinct points? Colinear Points A set of points is colinear if there is a single line through all of the points. (Note: Every set of two points is colinear.) Example Draw a set of three points which are colinear and another set of three points which are not colinear. Why Study Geometry? Formal Geometry Geometric Objects Conclusion Coplanarity Questions line those we asked about lines can be asked about planes as well. Example 1 How many planes are there through a single point? 2 How many planes are there through two points? 3 How many planes are there through three points? Coplanar Points A set of points is said to be coplanar if there is a plane containing all points in the set. Example Can you find a set of points which are not coplanar? Why Study Geometry? Formal Geometry Geometric Objects Conclusion From a Line To. Using the basic object of a line, we can define several new objects. Line Segment A line segment is a subset of the line which contains two points on the line, called endpoints, and all parts of the line between these two points. Rays A ray is a subset of a line that contains a specific point, called the endpoint, and all points on the line on one side of the endpoint. Example Draw an example of a line, line segment, and a ray and name each object using point names. Why Study Geometry? Formal Geometry Geometric Objects Conclusion Relationships Between Lines Two lines can have several different relationships to each other. Relationships Between Lines Two lines can have the following relationships: 1 Perpendicular The lines form right angles at their intersection. 2 Concurrent The lines intersect at a single point. 3 Parallel Lines in the same plane which do not intersect are called parallel. 4 Skew Lines Two lines which lie in different planes and do not intersect are called skew lines. Why Study Geometry? Formal Geometry Geometric Objects Conclusion Angles When two lines intersect they form an angle. An Angle An angle is the union of two rays which the same endpoint, called the vertex. Each ray is called a side of the ray. Another way to think of angles is as movement or change. One ray sweeps out a 45◦ angle. Example Draw and name two different angles using only two lines. Partitioning An angle partitions the plane into three pieces: the angle itself, the interior (the portion of the plan between the rays on the side of least change) and the exterior. Why Study Geometry? Formal Geometry Geometric Objects Conclusion Dealing with Angles There are several important tasks which we need to be able to accomplish with angles. Tasks with Angles Naming Angles Measuring Angles Classifying Angles Angles can be classified based on their measures. Angles are: Straight - if their measure is 180◦ Right - if their measure is 90◦ Obtuse - if their measure is between 180◦ and 90◦ Acute - if their measure is between 0◦ and 90◦ Reflex - if their measure is greater than 180◦ Why Study Geometry? Formal Geometry Geometric Objects Conclusion Classifying Angle Relationships Angles can also be classified by their relationship to each other. Angle Relationships Two angles are: Complementary - if their measures add to 90◦. Supplementary - if their measures add to 180◦. Adjacent - if they share a side and vertex. Vertical - if they share a vertex and sides on the same lines. Congruence vs. Equality Two angles are congruent if they have the same measure. To be equal the angles must made up of the same objects. Example Prove that vertical angles are congruent. Why Study Geometry? Formal Geometry Geometric Objects Conclusion Important Concepts Things to Remember from Section 8.1 1 Basics of Formal Geometry 2 Definition of Basic Geometric Objects 3 Relationships Between Basic Geometric Objects 4 Working with Lines and Angles.