Geometry Geo Earth metry Measure Geometry Earth Measure

Geometry • Geo • Earth • -metry • Measure • Geometry • “Earth Measure”

Definition • Geometry – Study of relationships between points, lines, planes and solids.

Ø The system was first written down by a Greek man named Euclid. Ø The Geometry that we will study this year is called Euclidean Geometry.

Category 1 Undefined Terms we accept without definition 3 types • Point • Line • Plane

Category 2 Defined Terms Definition • Terms that have a definition Many types that we will study during this course

Category 3 Postulates Definition- Euclid Today • Statements that we accept without proof • Had 5 postulates • We have many more

Undefined Term 1 • Point – Has no length, width or thickness – Indicates a position or place • Name a point with a capital letter Ex. A X Y

Undefined Term 2 • Line – Points grouped next to each other to form a straight line – Has length but no width or thickness

Naming a Line 1 • Using two points on the line – Pick any two in any order Ex. A BC B AB AC C Can be named in the following ways with any two of the letters. Notice the double arrows above the letters

Try This • Name the following line in three different ways Ex. Z X Y

Naming a line 2 • Use a lowercase cursive letter – Letter is not a point on the line – Letter stands for all the points on the line Ex. l Line l

Undefined Term 3 • Plane – Set of points that form an infinite surface

Naming a Plane • Use a capital letter in the corner of the plane – The letter is not a point – Use this figure to represent a plane – Remember the plane continues in all directions Plane A A

Defined Terms • Reminder – Defined terms are terms that have a definition – They are the second category in the Geometry System

Defined Terms cont. • Collinear points – Definition: all the points on a line – Any two points are collinear because they can both be on the same line – More than two points can be collinear, but do not have to be collinear

Ex. A E D C B ØAny two points are collinear in this diagram Øie: A and E; B and C; D and A; and so on ØThree points that are collinear are B C and D ØHowever Points A, B and D are not collinear ØIn this case A, B and D are called Noncollinear

Defined Terms cont. • Noncolinear – Definition: Points that are not contained on the same line – In this case A, C and D are called Noncollinear Ex. A D C

Defined Terms cont. • Coplanar – Definition: points all in the same plane • Noncoplanar – Definition points not in the same plane – Example: The Rectangular Prism

D C A B H G E Ex. F ØPoints A, B, C and D are coplanar ØPoints A, B, C and H are noncoplanar

• Ray – Definition: A part of a line that has one endpoint and all the points on one side of the endpoint – Could also be said that it is Half of a line • Naming a Ray – The endpoint of the ray must be the first letter followed by any letter(s) on the line. – The Name has a one way arrow above the letters

• Example Z Y This Ray can be called: X XY XZ XYZ X must always be first in this case because it is the endpoint

• Line Segment – Definition: A part of a line that has 2 endpoints and includes all of the points between the endpoints • Naming a Line Segment – Use the two endpoints in any order with a line above the name

• Example Z Y X This Line Segment can be called: XZ ZX XYZ ZYX X or Y must always be first or last in this case because they are the endpoints

Try This • Given ray QM – Is Z on this ray? – Is A on this ray? • Given Line Segment AZ – Is Q on this segment? – Is ray QZ on this segment A Z Q M

Solutions • Given ray QM – Is Z on this ray? • Yes – Is A on this ray? • No, point A is past the endpoint Q and is not on the ray QM • Given Line Segment AZ – Is Q on this segment? • Yes – Is ray QZ on this segment • No, ray QZ has points that are past the endpoint Z on the line segment AZ

• Postulates – Through any two points there is exactly one line A B C

• Postulates – Through any three non collinear points there is exactly one plane C B A

• Postulates – If two points lie in a plane then the line containing them also lies in that plane B C

• Postulates – If two planes intersect their intersection is a line A l B

• Postulate – If two lines intersect, then the intersect at exactly one point l P m