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# Unit 1 foundations of geometry

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• Ask: What do you suppose this picture is made of? (“color?” “dots?” “we call those dots, points”)
• Here are some more examples of art that use a collection of “points”. You can look up more works of art in these similar styles.
• Ask: What is this picture made of? (“Lines?” “Close, but what exactly is a line? Part of a line is called a line segment..”)
• Ask: “Can you find the ray?” then “what time(s) form opposite rays?”Reiterate that DIRECTION MATTERS on rays.
• Ask: “What are some planes? …”
• Start: Discovery Based Learning (ask “from our definitions, what is true? What can happen? How can this happen?” Think Pair Share activity, answer is student words)
• This should be a 5-10 minute group activity. If time permits.
• Ask: What is this picture made of, besides line segments? “Angles”
• Inquiry Based Learning: “What do these words mean?”Review: How do we label the points? Segments? Rays? Angles?
• Students are encouraged to paraphrase AFTER writing these exact steps. Extra examples on board with my GIANT yellow protractor.
• Inquiry Based Learning: “What do these words mean?”Review: How do we label the points? Segments? Rays? Angles?
• ### Transcript

• 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
• 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 Euclid’s 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. Let’s 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, France Sunday Afternoon on Isle de La Grande Jatte
• 16. Points an 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. Lines a 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
Opposite two rays that have a common
Ray endpoint and form a line.
Collinear points 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.
Coplanar points 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
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?
Theorem A statement that requires proofs and previous postulates. This technique utilizes deductive reasoning.
Postulate A statement is accepted as truth without proof. Also called an axiom.
• 38. Unit 1: Postulates and Theorems
(At the bottom of pg 7)
Summary Answer 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
Postulate Through any two points…
1-1-1
There is exactly one line.
Postulate Through any three non-collinear
1-1-2 points…
There is exactly one plane containing them.
Postulate If two points lie in a plane, then
1-1-3 the line containing those
points..
Lies in that plane.
Postulate If two lines intersect, then they
1-1-4 intersect…
In exactly one point.
Postulate If two planes intersect, then they
1-1-5 intersect…
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 Length pg. 9What does the Ruler Postulate mean and how does it define distance?
Parallel Lines Coplanar lines that do not intersect.
Perpendicular Lines that intersect to form a right
Lines angle
• 42. Unit 1: Distance and Length pg. 9What is the Ruler Postulate mean and how does it define distance?
Ruler Postulate Points 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)!!!
LENGTH To measure the LENGTH of 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)
Summary What 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. Angle A figure formed by two rays
with a common endpoint,
called a vertex
Written: A OR 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. Construction A method of creating a mathematically precise figure
using a compass and straight
edge, software, or paper
folding
How do I use 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.
• 51. Unit 1: All About ANGLESpg. 14
How can you name and classify an angle?
Protractor Postulate
When it’s 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
The measure of angle DEG = 115º, and the measure of angle = 48º. Find the measure of angle FEG.
F
D
E
G
• 55. BISECTOR
Ray KM bisects angle JKL, measure of angle JKM = (4x + 6)º, and the measure of angle MKL = (7x – 12)º. 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