INTRO-LECTURE- TO. MODERN-GEOMETRY.pptx

About the modern geometry.

1. MODERN GEOMETRY Chapter 1: Classical Euclidean Geometry

2. OVERVIEW: The origin of “geometry” comes from the Greek word “ geometrein” (geo means earth and metrein means to measure). Geometry was oiginally the science of measuring land. The study of Geometry is extremely ancient and has been carried on for many thousand of years, across all civilizations: Egypt, Babylonian, India, China, Greece, the Incas, etc. Geometry’s origins go back to approximately 3, 000 BC in ancient Egypt. Ancient Egyptians used an early form of geometry for a variety of purposes, including land surbeying, yramid construction and astronomy.

3. The Greeks were the first to establish the concept of proofs in a systematic way. According to the Rhind Papyrus, the Egyptians in 1800 BC had the approximation  = 3.1416. Egyptian Geometry was not a science in the Greek sense, but rather a collection of mathematical rules with no rationale or justification. In Arithmetic and mathematics, the Babylonians were far ahead of the Egyptians. They also knew the Pythagorean Theorem. Otto Neugebauer;s recent research has uncovered a previously undiscovered Babylonian Algebraic effect on Greek mathematics.

4. The Pythagorean Schools systematic foundation of Plane Geometry was brought to a concluion by the mathematician Hippocrates in the Elements around 400 BC. The Pythagoreans were never able to develop a proprotional theory that applied to irrational lengths. Eudoxus, whose theory was incorporated into Book V of Euclid’s Elements, later achieved this The next great advance in geometry was made buy Eulcid in 30o BC, when he wrote a book titled “Elements” Euclid’s Elements are a collection of 13 books that contain theorems, constructions, and geometrical proofs. .

5. Euclid also prensented an ideal axiomatic form in this text (now known as Euclidean Geometry). Euclidean geometry is the study of lines, angles, solid shapes, and figures using axioms and postulates to prove propositions using a small set of statements that are accepted as true.

6. MODERN GEOMETRY Undefined Terms

7. Undefined  terms in geometry refer to elements that, although often explained, do not have a formal definition. These elements serve as a foundation for other well-defined elements and theorems. The lack of a definition of terms like  point  and  line  do not make them less important or less concrete.  Undefined terms are concepts that are usually described through examples and visual representations for not having a formal description. Some examples are point, line, plane, and set . Each of these terms is of extreme importance for the construction of theorems and other concepts.

8. I n Euclidean geometry, undefined terms, which are arbitrary and could easily be replaced by other terms, normally include points, lines and planes; it would be possible to develop Euclidean geometry using such concepts as distance and angle is undefined. Definitions of new words involve the use of undefined terms. For example , when we say polygon, we mean that it is a plane figure bounded by a finite number of a line segments. However, from the definition itself, we must also define the other terms used: plane figure and line segment. If we define a line segment as a part of a line that is bounded by two endpoints. We have to define again the word line and endpoints. That’s why we have to stop somewhere, we have to have some undefined terms -terms that do not require definition.

9. POINT refers to the idea of an exact, fixed location.  A This is point A A point in geometry, can be defined as a dimensionless mark that represents a location in space. Its lack of dimensions refers to the absence of width, height, and depth of a point. This definition sounds counterintuitive because a point is visually presented by a small circle, which, being a geometric figure, has two dimensions. However, one must be aware that a visual representation usually brings an extrapolation of what an element actually is in theory.

10. A point indicates a location (or position) in space. A point has no dimension (actual size) A point has no length, no width, and no height (thickness) A point is usually named with a capital letter. In the coordinate plane, a point is named by an ordered pair, (x, y). The size of the dot drawn to represent a point makes no difference since points have no size and they simply represent a location.

11. Line LINE is a set of infinite many points arranged in a straight path which extends endlessly in opposite directions. Y X This is line XY A Line can also be described as infinitely long straight mark or band that goes on forever in both directions but have no width or height. A Line has no thickness. A line’s length extends in one dimension and goes one forever in both directions. A line has infinite length, zero width and zero height. A Line is assumed to be straight. (in Euclidean Geometry) A line is drawn with arrowheads on both ends.

12. Line A line is named by a single lowercase script letter, or by any two (or more) points which lie on the line. This is AB The thickness of a line makes no difference. Definition : Collinear points are points that lie on the same straight line. Postulate : One, and only one, straight line can be drawn through two dinstinct points.

13. Line PLANE is described as a flat surface with infinite length and width, but no thickness. A symbol of a plane in Geometry is usually trapezoid, to appear three-dimensional and understood to be infinitely wide and long. A single capital letter, or three pooints drawn on it, name the plane. A plane is named by a single letter or by three coplanar, but non-collinear points. While the diagram of a plane has edges, remember that the plane actually has no boundaries. Definition: Coplanar points are points that lie in the same plane

14. Line SET can be described as a collection of objects, in no particular order, that you are studying or mathematically manipulating. Sets can be all these things: Physical objects like angles, rays, triangles , or circles. Numbers, like all positive even integers; proper fractions; or decimals smaller than 0.001 3. Other sets, like set of all even numbers and the set of multiples of five; the set of acute angles and the set of all angles less than 15 . In geometry, we use sets to group numbers or items together to form a single unit, like all the triangles on a plane or all the straight angles on a coordinate grid. 

15. MODERN GEOMETRY Euclid’s First Four Postulates

16. Line Postulate 1: “ A straight line can be drawn from any point to another point.” This postulate states that at least one straight line passes through two distinct points but he did not mention that there cannot be more than one such line. Although throughout his work he has assumed there exists only a unique line passing through two points. This postulate can be extended to say that a unique (one and only one) straight line may be drawn between any two points.

17. Line Postulate 2: “ A terminated line can be further produced indefinitely.” In simple words what we call a line segment was defined as a terminated line by Euclid. Therefore this postulate means that we can extend a terminated line or a line segment in either direction to form a line. In the figure given, the line segment AB can be extended as shown to form a line. To produce a finite straight line continuously in a straight line or a terminated line (a line segmnt) can be produced indefinitely.

18. Line Postulate 3: “ A circle can be drawn with any center and any radius.” Any circle can be drawn from the end or start point of a circle and the diameter of the circle will be the length of the line segment . Given any straight line segment , a  circle, can be drawn having the segment as radius  and one endpoint as center. To describe a circle with any center and distance.

19. Line Postulate 4: “ All right angles are equal to one another.” All the right angles (i.e. angles whose measure is 90°) are always congruent to each other i.e. they are equal irrespective of the ir length of the sides or their orientations. Postulate 5 : “ Given a line L and a point P not on the line, exactly one line can be drawn through P which is parallel to L.

20. MODERN GEOMETRY Parallel Postulate or Euclid’s 5th Postulate

21. Line Euclid’s first four postlates have always been readily accepted by matehmaticians. The fifth postulate-the “Parallel Postulate” however, became highly controversial. The Fifth Postulate is often called the Parallel Postulate even though it does not specifically talk about parallel lines; it actually deals with the ideas of parallelism. The considerarion of alternatives to Euclid’s Parallel Postulate resulted in the development of non-Euclidean Geometry.

22. Line Postulate 5: “ If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines , if produced indefinitely, meet on that side on which the sum of angles is less than two right angles . ” If one line intersects two other lines, then can you tell whether these two lines are parallel or not? NOTE: If the sum of two interior angles is 180 , then it is a parallel line. In the given diagram, the sum of angle 1 and angle 2 is less than 180°, so lines n and m will meet on the side of angle 1 and angle 2.

23. Line Euclid says in his 5th postulate, if the sum of two angles is less than two right angles on the side, then two line would eventually meet on that side.

24. Line Moreover, if two parallel lines are cut by a transversal, then the corresponding angles are equal  . REMEMBER : A transversal line is a line that crosses or passes through two other lines. Corresponding angles are equal when two parallel lines are cut by a transversal. This postulate says that if  l  //  m , then 1. m  ∠1 =  m  ∠5 2. m  ∠2 =  m  ∠6 3. m  ∠3 =  m  ∠7 4. m  ∠4 =  m  ∠8

25. Line Alternate-Interior Angles are formed when a transversal passes through two lines. The angles that are formed on opposite sides of the transversal and inside the two lines are alternate-interior angles. Other examples of alternate-interior angles:

26. Line PROPOSITION: “If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another” PROOF: Let ST be a transversal cutting line AB and CD in such a way that angles BST and CTS are equal (pabeled a in the figure). Assume that AB and CD meet in a point P in the direction of B and D. Then in triangle SPT, the exterior angle CTS is equal to the interior opposite angle TSP. But this is impossible. It follows that AB and CD cannot meet in the direction of B and D. By similar argument, it can be shown that they cannot meet in the direction of A and C. Hence they are parallel.

27. Line PROPOSITION: “If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another” PROOF: Let the straight line EF falling on the two straight line AB and CD make the exterior angle EGB equal to the interior and opposite angle GHD, or the sum of the interior angle on the same side, namely BGH and GHD, equal to two right angles. AB is parallel to CD. Since the angle EGB equals the angle GHD and the angle EGB equals the angle AGH. Therefore, the angle AGH equals the angle GHD. And they are alternate. Therefore, AB is parallel to CD.

28. Line PROPOSITION: “If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two rright angles, the straight lines will be parallel to one another” PROOF : Continuation Next, since the sum of the angles BGH and GHD equals two right angles and the sum of the angle AGH and BGH also equals two right angles, therefore, the sum of angles AGH and BGH equals the sum pf the angles NGH anmd GHD. Subtract the angle BGH from each. Therefore, the remaining angle AGH equals the remaining angle GHD. And they ae alternate, therefore, AB is parallel to CD.

29. Line PROPOSITION: “If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two rright angles, the straight lines will be parallel to one another” PROOF : Continuation Therefore, if a straight line falling on two straight lines makes the exterior angle equal to the interiror and opposite angle on the same side or the sum of the interior angles on the same side qual to to right angles, the the straight line are parallel to one another.

30. Line PROPOSITION: “A straight line falling on parallel straight line make the alternate angles equal to one another, the exterior angle equal to the interior and opposite angl e , and the interior angles on the same side equal to two right angles.” PROOF : Let AB and CD be parallel lines cut in points S and T, respectively, by the transversal ST. Assume that angle BST is hreater than angle CTS. It folloes that the sam of angles BST and STD is greater than two right angles and consequently the sume of angles AST and CTS is less than two right angles. Then, by Postulate 5, AB and CD must meet.

31. Line PROPOSITION: “A straight line falling on parallel straight line make the akternate angles equal to one another, the exterior angle equal to the interior and opposite nagle, and the interior angles on the same side eul to two right angles.” PROOF : Continuation We conclude that angle BST cannot be greater than angle CTS. In a simialr way , it can be shown that angle CTS cannot be greater than angle BST. The two angles must be equal and the first part of the proposition is proved. The remaining parts are then easily verified.

32. MODERN GEOMETRY Attempts to Prove the Parallel Postulate

33. Line Given: Point P not on line K. Let Q be the foot of the perpendicular form P to K. Let m be the line through P perpendicular to line PQ. Let m is parallel to k. Let n be any line through P disticnt from m and line PQ. Let ray PR be a ray of n between ray PQ and a ray of m emanating from P. There is a point A between P and Q.

34. Line Given: Point P not on line K. Let B be the unique point such that Q is between A and B and AQ is congrunet to QB . Let S be the foot of the perpendicular from A to n. Let C be the unique point such that S is between A and C and AS is congruent to SC. There is a unique circle G passing through A, B and C.

35. Line Given: Point P not on line K. K is the perpendicular bisector of AB, and n is the perpendicular bisector of AC. K and n meet at the center of G. The parallel postulate have been proven.

36. MODERN GEOMETRY Substitutes for Euclid’s Fifth Postulate

37. Line Playfair’s Axiom: “ Through a given point, not on a given line, exactly one line can be drawn parallel to the given line.” Playfair’s Axiom is equivalent to the 5th Postulate in the sense that it can be deducted from Euclid’s five postulates and common notios, while conversely, the 5th Postulate can be deduced from Playfair;s Axiom together with the common notions and first tour postulates. The Angle-Sum of a Triangle: “ A second alternative for the 5th Postulate is the familiar Theorm: (The sum of the three angles of a triangle is always equal to two right angles.)” This is a consequence of Playfair’s Axiom, and hence of the 5th Postulate, is well known.

38. Line The Existence of Similar Figures The following statement is also equivaent to the 5th Postulate and may be susbtituted for it, leading to the same consequences: “There exists a pair of similar triangles, i.e., triangles which are not congruent , but have the three angles of one equal, respectively, to the three angles of the other.” Equidistant Straight Lines Another noteworthy substitute is the following: “There exists a pair of straight lines everywhere equally distant from one another.” Once the 5th Postulate is adopted, this statement follows, for them all parallels have this property of being everyhere equally distant. F the above statement is postulated, we can easily deduce the 5th Postulate by first provig that there exists a triangl with the sum of its angles equal to two right angles.

39. Line