Intro to Geometry

INTRODUCTION TO GEOMETRY

GEOMETRY: A HISTORY

• Said to have been invented over 4,000 years ago by Egyptian pharaoh Sesostris to help keep track of land ownership and tax its owners• “Geo” means earth, “metria” means to measure; therefore, Geometry literally means to measure the

Earth• Greek philosophers built upon the very practical mathematics they learned from the Egyptians

and Babylonians to create a more abstract and general way of thinking of Earth measures• The assumptions they developed were minimal which allowed everything else to follow from

these basic assumptions• Many of the original texts were lost during the fall of the Roman empire, but the teachings

remained prevalent in the Islamic nations.

EUCLIDEAN GEOMETRY

• Euclid’s, The Elements, summarized Greek geometry. It is the basis of most Western mathematics, science, and philosophy

• Euclidean geometry dates back to approximately 400 BC• Older than algebra and Calculus!

• Many still believe Euclidean geometry is the best introduction to analytic thinking• We will follow the basic thinking developed by Euclid and attempt to make clear

and distinguish between:• What we have assumed to be true, and cannot prove• What follows from what we have previously assumed or proven

• Essentially, we will always question every idea presented.

THINK OUTSIDE THE BOX…

POINTS, LINES, & PLANESUnit 1

EUCLID’S GEOMETRY

• Euclid’s assumptions are referred to as axioms, postulates, and definitions

• Axioms are very general ideas; postulates and definitions refer to specific ideas

• Definitions are words or terms that have agreed upon meaning; they cannot be proven or derived

• Major ideas which are proven are called theorems• Ideas that follow from a theorem are corollaries• Euclid referred to his five axioms as “Common Understandings”

COMMON UNDERSTANDINGS

• Axiom 1: Things that are equal to the same thing are also equal to each other• Axiom 2: If equals are added to equals, the whole are equal• Axiom 3: If equals are subtracted from equals, the remainders are equal• Axiom 4: Things which coincide with one another are equal to one another• Axiom 5: The whole is greater than the part

THE UNDEFINED TERMS

Point• Simplest figure in Geometry• Everything else consists of

points• Used for location

• Does not have a “size”• Represented by a dot

• Labeled with an uppercase non-cursive letter

Line• Extends infinitely in two directions• Made of an infinite amount of

points • Does not have “thickness”• Represented with 2 arrows in

opposite directions• Labeled with any two points on

the line OR a single lowercase cursive letter

Plane• Extends infinitely in all directions

• No edges• No thickness

• Need at least three points to create a plane

• Represented with a parallelogram• Labeled with at least three points

OR a single uppercase cursive letter

MORE IMPORTANT DEFINITIONS

• Collinear• Points of the same line

• Non-Collinear• Points not on the same line

• Coplanar• Figures (points or lines) on the same plane

• Non-Coplanar• Figures not on the same pane

• Space• A boundless, three-dimensional set of all points

POSTULATES

• 1: Through any two points exists exactly one line

• 2: Through any three non-collinear points exists exactly one plane

INTERSECTIONS

• Two Lines• A Point

• Two Planes• A Line

• A Lines and a Plane• A Point (line goes through plane)