MATH 113 Section 8.1: Basic Geometrymath.wallawalla.edu/~duncjo/courses/math113/winter08/... · · 2008-01-28Formal Geometry Geometric Objects Conclusion The Study of Shapes One

Why Study Geometry? Formal Geometry Geometric Objects Conclusion

MATH 113Section 8.1: Basic Geometry

Prof. Jonathan Duncan

Walla Walla University

Winter Quarter, 2008


Outline

1 Why Study Geometry?

2 Formal Geometry

3 Geometric Objects

4 Conclusion


Geometry in History

Our modern concept of geometry started more than 2000 yearsago with the Greeks.

Plato’s Academy

To 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.


The Study of Shapes

One 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 maps

building plans

schematics for drawings

artistic portrayals

others?


Tetris and Mathematical Thinking

Geometry is also useful in stimulating mathematics thinking. Takefor example the game of Tetris.

Mathematical Thinking and Tetris

Defining 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?


Two Types of Geometry

Traditionally, geometry in education can be divided into twodistinct types.

Types of Geometry

Formal 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.


Euclid’s Postulates

We can not prove everything! We must have some starting pointto any formal system.

Euclid’s Postulates

One of the most famous books 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.

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 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.


Euclid’s Fifth Postulate

To 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.


Basic Objects in Geometry

We now turn to the more conceptual questions in geometry. Thoseinvolving basic objects and their relationships.

Basic Geometric Objects

The 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 location

Lineslines 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.


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 thepoints. (Note: Every set of two points is colinear.)

Example

Draw a set of three points which are colinear and another set ofthree points which are not colinear.


Coplanarity

Questions line those we asked about lines can be asked aboutplanes 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 containingall points in the set.

Example

Can you find a set of points which are not coplanar?


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 pointson the line, called endpoints, and all parts of the line betweenthese two points.

Rays

A 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.

Example

Draw an example of a line, line segment, and a ray and name eachobject using point names.


Relationships Between Lines

Two lines can have several different relationships to each other.

Relationships Between Lines

Two lines can have the following relationships:

1 PerpendicularThe lines form right angles at their intersection.

2 ConcurrentThe lines intersect at a single point.

3 ParallelLines in the same plane which do not intersect are calledparallel.

4 Skew LinesTwo lines which lie in different planes and do not intersect arecalled skew lines.


Angles

When two lines intersect they form an angle.

An Angle

An angle is the union of two rays which the same endpoint, calledthe vertex. Each ray is called a side of the ray.

Another way to think of angles is as movement or change. One raysweeps 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, theinterior (the portion of the plan between the rays on the side ofleast change) and the exterior.


Dealing with Angles

There are several important tasks which we need to be able toaccomplish 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◦


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 equalthe angles must made up of the same objects.

Example

Prove that vertical angles are congruent.


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

Documents

MATH 113 Section 8.1: Basic Geometrymath.wallawalla.edu/~duncjo/courses/math113/winter08/... · · 2008-01-28Formal Geometry Geometric Objects Conclusion The Study of Shapes One