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- 1. Lesson BNHS - No. 1 Gawaran Euclidean Geometry Fundamental Ideas in Geometry Alexander Marzonia Morron Jr Mathematics Educator
- 2. Geometry The word "geometry" comesfrom two Greek wordsgeo and metronMeaning "earth measuring."
- 3. Geometry is the visual study of shapes, sizes,patterns, and positions. It occurred in all cultures, through at least oneof these five strands of human activities:1. building/structures (building/repairing a house, laying out a garden, making a kite, …)2. machines/motion (using a pry-bar, riding a bike, sawing a board, swinging, …)3. navigating/star-gazing (How do I get from here to there?, using maps, …)
- 4. Geometry is the visual study of shapes, sizes,patterns, and positions. It occurred in all cultures, through at least oneof these five strands of human activities:4. Art/patterns (designs, symmetries, representations, …)5. Measurement (How big is it?, How far is it?, ...)
- 5. Millau Bridge Millenium Park Sir Norman Foster FallingwatersFrank Lloyd Wright Frank Lloyd Wright Point, Lines, and Planes Components of Geometry
- 6. Three undefined terms:Points, Lines, and Planes. (P L P)• We can give descriptions of these three terms.• We also use these terms to help us write definitions of other terms such as segment, or ray
- 7. WHY THESE ARE CALLED UNDEFINED? It is because they can only bedefined circularly. Which means in terms of eachother or in terms of themselves.
- 8. POINT A point is usually represented by adot on a piece of paper, a pinhole, andthe tip of a sharp needle.It is so small that it can never bemeasured because it has no dimensionsat all.
- 9. - It shows location.- It is named by one capital letter. A B CA, B, and C are POINTS
- 10. The pictures shown above represent a line.
- 11. LINESA line is always straight and travels forever(INFINITE) in two directions.A line may be represented by the edge of a ruleror a piece of stretched thread.has no width BUT IT HAS length( which isinfinite).
- 12. ILLUSTRATION A B C The opposite arrow indicates thatit extends indefinitely in oppositedirection.
- 13. LINES CAN BE NAMED IN 2 WAYS:1. BY TWO POINTS. HENCE, TWO POINTS DETERMINE ALINE.FOR EXAMPLE, IN THE FIGURE,Line AB or AB. Or Line BA orBA A B CLine AC or AC. Or Line CA orCA
- 14. 2. LINES ARE ALSO NAMED WITHLOWERCASE LETTERS OR A SINGLELOWER CASE LETTER. m A B C The line above can benamed also as line m insteadof line AB or line AC.
- 15. COLLINEAR POINTS & NON- COLLINEAR POINTS A B CA, B, and C are COLLINEARPOINTS
- 16. COLLINEAR POINTS & NON- COLLINEAR POINTS Q P RP, Q, and R are NON-COLLINEAR POINTS
- 17. COLLINEAR POINTS-Points that lie on the same line.NONCOLLINEAR POINTS- Points that do not lie on the sameline.
- 18. ExamplesCOLLINEAR POINTS T A L- B, A, L, K B K- S, M, K D H- Q, M, ? M- T, ? , D S Q
- 19. ExamplesNONCOLLINEAR POINTS T A L- B, A, M, K B K- D, M, K D H- Q, M, ? M- T, ? , D S Q
- 20. THE FLOOR
- 21. THE WALL AND THE CEILING
- 22. PICTURE FRAME
- 23. PLANE A plane is often represented by a FLATSURFACE. These plane surfaces are used toconnect any two or more points on astraight line. The walls of the classroom, the top of atable, and the surface of the chalkboardsare examples of plane.
- 24. -A plane has nothickness andedges.- has lengthand width. β
- 25. In naming,-Named by 3distinct pointswhich are notcollinear. D-For example,plane BDC. C B
- 26. or-Single greekalphabet.-For example,Plane β β
- 27. HOW MUCH DO YOU KNOW…• What are the basic terms of geometry?
- 28. QUIZ• Determine the undefined term suggested by each of the following.• 1. the tip of a pencil• 2. the top of a coffee table• 3. telephone wires• 4. the wall of a room• 5. the surface of the page of a book
- 29. QUIZ• Determine the undefined term suggested by each of the following.• 6. the ruler’s edge• 7. the tip of a needle• 8. a window pane• 9. the floor of your bedroom• 10. the string on a guitar•
- 30. • Euclidean Geometry - History:• Euclid was the first mathematician of his time to use the axiomatic method.• Euclid’s axiomatic system consisted of four parts: undefined terms, which lead to definitions, which lead to postulates (also called axioms, which are accepted as true), which lead to theorems (which are proven by deductive reasoning). 33
- 31. • Euclidean Geometry – History:• The undefined terms on which Euclid based his system were point, line, and plane.• Point – a location in space• Line – a straight arrangement of points• Plane – a two dimensional surface that extends infinitely in both directions ( i.e., a table top )• Euclid used the undefined terms to introduce certain definitions as they were needed in his axiomatic system. 34
- 32. 3 Undefined TermsA point is like a star in the night sky. It is a little spec in a large endless sky. However, unlikestarts, geometric points have no size. Think of them as being so small that they take up zeroamount of space.A line is like the edge of a ruler, that never ends. Take a ruler and draw a line - now imagine ifthat line kept going straight forever. The line you have is thick enough for you to see, but youneed to imagine that your line is so thin that you cant see it - it has no thickness at all. That isa geometric line.A plane is a flat piece of land (like a football field) that extends forever. Imagine that you canpick that football field up, and put it anywhere in the air that you like. You can even turn itside ways, or diagonally. A plane is just a flat edge (like a piece of paper) that has nothickness (just like the line) and extends forever.
- 33. UNDEFINED TERMS• Undefined terms are terms that are accepted without definition.• The undefined terms we will use are: set, point, line and plane.
- 34. Aim: What are the undefined terms? Do Now: 1) What is a point? 2) What is a line? 3) What is a plane? The most fundamental concepts in geometry do not have precise definitions but, we understand their meanings intuitively. We say these concepts are undefined. Geometry Leeson: Undefined Terms, Lines, 37 Line Segments
- 35. A point indicates a position or locationPoint: in space. . Y P . A(2, 6) X Points are named using capital letters and/or coordinates. Geometry Leeson: Undefined 38 Terms, Lines, Line Segments
- 36. A line is an infinite set of adjacent points.Line: Ex: Curved line Ex: Straight line Naming a Line: a) Two points on the line: A B , A C , B A , B C , etc ... A B C b) Single lowercase letter m Geometry Leeson: Undefined Terms, Lines, 39 Line Segments
- 37. Plane: A plane is a set of points that forms a completely flat surface. Naming a Plane: a) Three points on the plane: B • Plane ABC C • A • b) Single uppercase letter: Plane R R Geometry Leeson: Undefined Terms, Lines, 40 Line Segments
- 38. Def: Collinear Points: A collinear set of points is a set of points all of which lie on the same straight line. E A B C D •Points A, B, C and D are collinear. •Points A, E and C are not collinear. Geometry Leeson: Undefined Terms, Lines, 41 Line Segments
- 39. Def: Line Segment: A B A line segment is the set of two points on a line called endpoints, and all points on the line between the endpoints. Naming a Line Segment: Use the names of the endpoints. A B “Line segment” A B is part of “Line” AB Geometry Leeson: Undefined Terms, Lines, 42 Line Segments

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