1. Euclidean vsNon-Euclidian Geometry
Ms. Hayde Rivas

2. Bell Ringer
Take out Homework and put on top of desk
Parent form
article
Pick up a School Map
Correct your quiz (Answers are posted on walls)
3. Taxi-Cab
http://www.learner.org/teacherslab/math/geometry/shape/taxicab/index.html
4. Parallel
Computer designs run programs simultaneously
http://www.cse.psu.edu/~teranish/ri_02.html
Extra Europa Parallel
By:DietmarTollerian
5. Spherical
6. Hyperbolic
7. Summarize the Article
What is Non-Euclidean Geometry?
By Joel Castellanos
8. 1.1 Euclidean Geometry
Euclid is a Greek mathematician.
Euclid lived in 300 B.C.
Euclid wrote a book The Elements
In high school, we study The Elements which is Euclids 2000 year old book.
Greeks used Euclidean Geometry to design buildings, predict locations and survey land.
9. 1.2 Non-Euclidean Geometry
Any geometry different from Euclidean geometry.
Each system of geometry has different definitions, postulates and proofs.
Spherical geometry and hyperbolic geometry are the most common Non-Euclidean Geometry.
The essential difference between Euclidean geometry and non-Euclidean is the nature of parallel lines.
10. 1.3 Spherical Geometry
Spherical Geometry is geometry on the surface of a sphere.
Lines are the shortest distance between two points.
All longitude lines are great circles.
Spherical Geometry is used by pilots and ship captains.
11. 1.4 Hyperbolic Geometry
Hyperbolic geometry is the geometry of a curved space.
Same proofs and theorems as Euclidean geometry but from a different perspective.
12. Lets Summarize..again
Euclidean Geometry
Non-Euclidean Geometry
Euclid was a Greek mathematician.
The Elements was written by Euclid sometime 300 BC
The concepts studied in Geometry today.
Non-Euclidean geometry is any geometry different from Euclidean geometry.
Three types of Non-Euclidean geometry are Taxi-Cab, Spherical geometry and Hyperbolic geometry.
This applications are used for maps, global traveling or space traveling.
13. Point, Line, Plane line segment or Ray
State Whether It is a..
14. Unit 1:Points, Lines and Planes pg. 5
What are the undefined terms of Euclidean Geometry?
Points
15. George Seurat (1859-1891) Paris, FranceSunday Afternoon on Isle de La Grande Jatte
16. Pointsan object or location in space that has no size (no length, no width)
Art styles:neo-impressionism, pointillism, divisionism
Charles Angrand (1854 -1926) , NormandyCouple in the Street
Henri-Edmond Cross(1909)The Church of Santa Maria degli Angeli near Assisi
17. Linesa straight path (a collection
of points) that has no thick-ness and extends forever.
Written: AB
Endpoint a point at one end of a segment or the starting point
of a ray.
Line a straight path (a collection
Segment of points) that has no thick- nessand two endpoints.
Written: AB
Pablo Picasso (1881-1973) Malaga, Spain
The Guitar Player
18. Ray part of a line that has one
endpoint and extends forever
in one direction.
Written: AB
Oppositetwo rays that have a common
Rayendpoint and form a line.
Collinearpoints that are on the same
line.A
B
C
Non-not collinear
Collinear
A,B, C as a group
19. Salvador Dali (1904-1989) Figueres, Spain
Skull of Zarbaran
Plane a flat surface that has no
thickness and extends forever.
Coplanarpoints that are on the same
plane
Non- not coplanar
Coplanar
Art styles:cubism
20. Unit 1:Points, Lines and Planes pg. 6
Summary The three undefined terms of Euclidean geometry are___________, ________________ and ____________.
21. A TAUT PIECE OF THREAD
Line Segment
22. A KNOT ON A PIECE OF THREAD
Point
23. A PIECE OF CLOTH
Plane
24. THE WALLS IN YOUR CLASSROOM
Plane
25. A CORNER OF A ROOM
Point
Corner
26. THE BLUE RULES ON YOUR NOTEBOOK PAPER
Line Segments
27. YOUR DESKTOP
Plane
28. EACH COLOR DOT, OR PIXEL, ON A VIDEO GAME SCREEN
Point
29. A TELECOMMUNICATIONS BEAM TO A SATELITE IN SPACE
Ray
30. A CREASE IN A FOLDED SHEET OF WRAPPING PAPER
Line Segment
31. A SHOOTING STAR
Ray
32. THE STARS IN THE SKY
Point
33. Y=MX+B
Line
34. A CHOCALATE CHIP PANCAKE
Plane
35. THE CHOCOLATE CHIPS IN THE PANCAKE
Point
36. Remember
A point is an exact location without a defined shape or size
A Line goes on forever
A Plane is a flat surface
A Ray has ONE endpoint
A Segment has TWO endpoints
37. Unit 1:Postulates and Theorems pg. 7
What are the defined terms of Euclidean Geometry?
TheoremA statement that requires proofs and previous postulates.This technique utilizes deductive reasoning.
PostulateA statement is accepted as truth without proof. Also called an axiom.
38. Unit 1:Postulates and Theorems
(At the bottom of pg 7)
SummaryAnswer Essential Question in Complete Sentences.What are the defined terms of Euclidean Geometry?
The defined terms of Euclidean Geometry ____________ and ____________ . The first term is defined as ____________ . The second term is defined as ____________ .
39. Glue the POSTULATE sheet so that it is able to flap open. Cut along the dotted lines. PG 8
PostulateThrough any two points
1-1-1
There is exactly one line.
PostulateThrough any three non-collinear
1-1-2 points
There is exactly one plane containing them.
PostulateIf two points lie in a plane, then
1-1-3the line containing those
points..
Lies in that plane.
PostulateIf two lines intersect, then they
1-1-4intersect
In exactly one point.
PostulateIf two planes intersect, then they
1-1-5intersect
In exactly one line.
40. Activity 1
Create a picture using only points
Create a picture using line segments (label endpoints)
Create a picture for each postulate.
41. Unit 1:Distance and Lengthpg. 9What does the Ruler Postulate mean and how does it define distance?
Parallel LinesCoplanar lines that do not intersect.
Perpendicular Lines that intersect to form a right
Lines angle
42. Unit 1:Distance and Lengthpg. 9What is the Ruler Postulate mean and how does it define distance?
Ruler PostulatePoints on a line can be paired with real numbers and distance between the two points can be found by finding the absolute value of the difference between the numbers.
REMEMBER:All distance must be Positive (In GEOMETRY)!!!
LENGTHTo measure the LENGTHof a
Distance (on a number line) segment, you can use a number line to find the DISTANCE between the two endpoints, or you can use the formula.
43. Unit 1:Postulates and Theorems
(At the bottom of pg 9)
SummaryWhat does the Ruler Postulate and how does it define distance?
The Ruler Postulate states ______________.It defines distance as _____________.
44. Ruler Postulate Examples pg 10
45. Segment Addition Postulatepg. 11
46. Segment Addition Postulate pg 12
47. Unit 1:All About ANGLESpg. 1
How can you name and classify an angle?
Angle
48. AngleA figure formed by two rays
with a common endpoint,
called a vertex
Written: AOR BAC
Side
Vertex
Side
Ray part of a line that has one
(Sides) endpoint and extends forever
in one direction.
Written: AB
Vertex the common endpoint of the
(End-sides of an angle
point)
Pablo Picasso (1881-1973) Malaga, Spain
The Guitar Player
49. Interior of
an Angle
Exterior of an Angle
Measure
Of an Angle
Congruent Angles
Degree
The set of all points between the sides of an angle
A
The set of all points outside an angle
B
Angles are measured in degrees.
C
Angles with equal measures.
of a complete circle
50. ConstructionA method of creating a mathematically precise figure
using a compass and straight
edge, software, or paper
folding
How do I use1.) Line up the center hole of A protractor?the protractor with the point
or vertex (corner)
2.) Line up a side (line) with
the straight edge of the
protractor
3.) Read the number that is
written on the protractor at
the point of intersection
(start from zero and count
up). This is the measure of
the angle in degrees.
51. Unit 1:All About ANGLESpg. 14
How can you name and classify an angle?
Protractor Postulate
When its a straight line the angles sum up to be
52. Protractor Postulate
Measure the Angles
How to Use a Protractor
1.) Line up the center hole of A protractor the protractor with the point or vertex (corner)
2.) Line up a side (line) with the straight edge of the protractor
3.) Read the number that is written on the protractor at the point of intersection (start from zero and count up). This is the measure of the angle in degrees.
Name a right angle and an acute angle:

Right = ________Acute = __________

What is the measure of the only obtuse angle shown?

Obtuse measure = ________
53. An angle that measures greater than 0 AND less than 90
An angle that measures EXACTLY 90
An angle that measures greater than 90 AND less than 180
An angle that measures EXACTLY 180
Acute
Right
Obtuse
Straight
A
B
C
54. ANGLE ADDITION POSTULATE
The measure of angle DEG = 115, and the measure of angle = 48.Find the measure of angle FEG.
F
D
E
G
55. BISECTOR
Ray KM bisects angle JKL, measure of angle JKM = (4x + 6), and the measure of angle MKL = (7x 12).Find the measure of angle JKL.
J
M
K
L
56. 57. Theorems
Congruent
Congruent
Congruent
58. Vertical Angles
59. Complementary Angles
60. Sources
Geometry, Holt
Sarah Gorena
C-Scope