Summarize the Article “What is Non-Euclidean Geometry?” By Joel Castellanos
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.
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.
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.
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.
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.
Point, Line, Plane line segment or Ray State Whether It is a…..
Unit 1: Points, Lines and Planes pg. 5 What are the undefined terms of Euclidean Geometry? Points
George Seurat (1859-1891) – Paris, France Sunday Afternoon on Isle de La Grande Jatte
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
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
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
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
Unit 1: Points, Lines and Planes pg. 6 Summary The three undefined terms of Euclidean geometry are___________, ________________ and ____________.
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
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.
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 ____________ .
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.
Activity 1 Create a picture using only points Create a picture using line segments (label endpoints) Create a picture for each postulate.
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
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.
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 _____________.
Unit 1: All About ANGLESpg. 1 How can you name and classify an angle? Angle
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
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
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.
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
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 = ________°
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
ANGLE ADDITION POSTULATE The measure of angle DEG = 115º, and the measure of angle = 48º. Find the measure of angle FEG. F D E G
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