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# College Geometry MATH 121

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This 6 page Class Notes was uploaded by Mrs. Benny Kohler on Monday October 5, 2015. The Class Notes belongs to MATH 121 at California State University - Sacramento taught by Jianyuan Zhong in Fall. Since its upload, it has received 33 views. For similar materials see /class/218799/math-121-california-state-university-sacramento in Mathematics (M) at California State University - Sacramento.

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Date Created: 10/05/15

Math 121 Chapter 2 Notes Introduction In this chapter welll informally assess Euclidls Elements as a for mal system and compare it with 1 Hilbertls Axioms for Euclidean Geometry7 and 2 the SMSG Postulates for Euclidean Geometry 1 22 EUCLID7S GEOMETRY AND EUCLID7S ELEMENTS The Greek mathematicians of Euclidls time thought of geometry as an abstract model of the world in which they lived The notions of point7 line7 plane7 and so on were meant to be consistent with human perception The postulates and axioms were mostly common sense and could not be questioned Euclid s Geometry Examples of two de nitions 1 A point is that which has no part 2 A line is breadthless length Postulates 1 To draw a straight line from any point to any point 2 To produce a straight line continuously in a straight line 3 To describe a circle with any center and radius 4 That all right angles are equal to one another 5 That7 if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles7 the two straight lines7 if produced inde nitely7 meet on that side on which are the angles less than two right angles Other translations according to Euclidean and Non Euclidean Ge ometries by Marvin Jay Greenberg7 3rd edition Euclid s Postulate I For every point P and for every point Q not equal to P7 there exists a unique line I passes through P and Q lnformally Two points determines a unique line Euclid s Postulate II For every segment AB and for every seg ment CD7 there exists a unique point E such that B is between A and E and segment CD is congruent to segment BE lnformally Any segment AB can be extended by a segment BE congruent to a given segment CD Euclid s Postulate III For every point 0 and for every point A not equal to 07 there exists a circle with center 0 and radius OA Euclid s Postulate IV All right angles are congruent to each other Euclid s Postulate V The Parallel Postulate Playfair s Pos tulate For every line I and for every point P that does not lie on l7 there exists a unique line m through P that is parallel to l 1 Challenging Euclid s Geometry Euclid7s presentation of plane geometry was awed on at least three counts 1 Failure to recognize the need for unde ned terms 2 Use of subtle but unstated postulates in the proof of theorem Example The postulate there exist at least three points that are not on the same line This was used in proof as unsaid assumption Unless the existence of points is explicitly postulated7 the existence of points may not be assumed Geometries lacking an existence postulate could be vacuous When proposing a geometric system7 modern geometers always postulate the existence of points in some way to guarantee that the theorems apply to a nontrivial model 3 Reliance on diagrams to guide the logic in the construction of proofs As we know7 by diagram is not allowed as a justi cation in a proof Example7 see the danger of relying on diagrams in proving all trian gles are isosceles on page 42 Several equivalent statements of Euclid s fth postulate 1 Through a given point only one line can be drawn that is parallel to a given line 2 There exists at least one triangle in which the sum ofthe measures of the interior angles is 180 3 Parallel lines are everywhere equidistant 4 There exist two triangles that are similar but not congruent 5 There exist two straight lines that are equidistant at three different points 6 Every triangle can be circumscribed 7 The sum of the measures of the interior angles is the same for all triangles 8 Rectangles can be constructed using a compass and a straightedge Homework Problems 227 1 27 4 8 on page 45 47 2 24 HILBERT7S AXIOMS FOR EUCLIDEAN GEOMETRY Since 18007 several mathematicians took on the task of constructing an axiom set that would result in the theorems of Euclid and that would comply with modern standards The most well known of these works7 Grundlagen der Geametrie Foundations of Geometry published in 1899 by a German mathematician7 David Hilbert David Hilbert7s Axioms Unde ned terms point7 line7 plane7 on incidence7 between7 con gruence 3 Hilbert divide his axioms into ve groups axioms of connection incidence axioms of order axioms of congruence axioms of parallels and axioms of continuity De nition Line Segment E Given two points A and B The line segment E is the set whose members are the points A and B and all points that lie on the line A B and are between A and B The two points A and B are called the endpoints of the segment AB De nition Circle Given two points 0 and A The set of all points P such that segment OP is congruent to segment 0A is called a circle with O as center and each of the segments OP is called a radius of the circle E 7gt De nition Ray AB Given two points A and B The ray AB is the set of points lying on thelhe A B those points belong to the segment AB and all points 0 on AB such that B is between A and C The ray A Bgt is said to emanate from the vertex A and to be part of line De nition Rays A Bgt and KC are opposite if they are distinct if they emanate from the same point A and if they are part of the same lt gt lt gt line AB AC De nition An angle with vertex A is a point A together with two distinct nonopposite rays A Bgt and E called the sides of the angle emanating from A De nition If two angles lt BAD and lt CAD have a common side E 7gt 7gt AD and the other two sides AB and AC form opposite rays the angles are supplements of each other or supplementopy angles De nition An angle lt BAD is a right angle if it is congruent to its supplementary angle Group I Axioms of Connection Incidence I 1 Through any two distinct points A B there is always a line in 12 Through any two distinct points A B there is not more than one line m lt We denote the unique line that passes through A and B by AB 13 On every line there exist at least two distinct points A B there exist at least three points that are not on the same line 14 Through any three points not on the same line there is one and only one plane Theorem Two distinct lines canlt intersect in more than one point Group H Axioms of OTdET II l If B is a point between A and 0 denote by A 7 B 7 C then A B C are distinct points on the same line and C 7 B 7 A II 2 For any two distinct points A and 0 there is at least one point on line A C such that A 7 C 7 B 113 If A B C are three points on the same line then exactly one is between the other two II 4 Let ABC be three points not on the same line let m be a line in the plane containing ABC that doesnt contain any of the three points Then if m contains a point of the segment E it will also contain a point of the segment E or a point of the segment W Theorem Every line contains an in nity of points Theorem If A and B are points then there always exists a third point C such that A 7 C 7 B Group III Axlams 0f Cangmence III1 If A and B are two distinct points on line a and if A is a point on the same or another line a then its always possible to nd a point B on a given side of the line a through A such that segment E is congruent to segment A B III2 For a segment A B and a segment A B are congruent to the same segment E then the segment A B and the segment A B are congruent Two segments congruent to a third segment are congruent to each other III3 additivity of segments On line a let M and E be two segments that except for B have no point in common Filrthermore on the same or another line a let A B and BC be two segments that except for B have no point in common In the case if E E A B and W 2 BCquot then E 2 AG III4 lf lt ABC is and angle and if f0 is a ray then there a lt gt is exactly one ray B Al on each side of BC such that lt ABC is congruent to lt ABC In particular each angle is congruent to itself III5 If for two triangles AABC and AA B C the congruences E g A B E g AC and lt BAC lt B A C are valid then lt ABC lt A B O Graup IV Axlams 0f Parallels Let a be any line and A a point not on it Then there is at most one line in the plane determined by a and A that passes through A and doesnt intersect a Group V Axlams 0f Cantmm39ty Vl Axiom of Archimedes If E and W are any segments then there exists a number n such that 71 copies of W constructed continuously from A along ray A Bgt will pass beyond the point B V2 Postulate of line Completeness An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as 5 the fundamental properties of line order and congruence that follow from Axiom I through 111 and V is impossible According to Euclidean and Non Euclidean Geometries by Marvin Jay Greenberg 3rd edition we have others such as Circular Continuity Principle If a circle 7 has one point inside and one point outside another circle 7 then the two circles intersect in two points Elementary Continuity Principle If one endpoint of a segment is inside a circle and the other outside then the segment intersects the circle Many theorems and proofs are to come in chapters 3 and 4 Homework 24 681213 3 26 THE SMSG POSTULATES FOR EUCLIDEAN GEOMETRY Go over historic background Homework write the SMSG Postulates on a piece of paper and turn it in hint Appendix D 4 27 NON EUCLIDEAN GEOMETRIES Hyperbolic Parallel Postulate There exist a line I and a point P such that at least two distinct lines pass through P that are parallel to l on page 74 Several equivalent statements of Hyperbolic parallel postu late 1 Through a given point can be drawn an in nite number of parallels to a given line 2 There exist no triangles in which the sum of the angle measures is 180 3 There exist no triangles that are similar but not congruent 4 There exist no lines that are everywhere equidistant 5 There exist triangles that cant be circumscribed 6 The sum of the measures of the interior angles of triangles varies among triangles and is always less than 180 7 Rectangles do not exist 8 There is an upper limit to the area of a triangle 9 the larger a trianglels area the smaller its angle sum 10 The distance between certain pairs of parallel lines approaches zero in one direction and becomes in nite in the other direction 11 If two parallel lines are crossed by a transversal the alternate interior angles may not be congruent Elliptic Parallel Postulate if I is a line and P is a point not on I then no line through P is parallel to Z In Hyperbolic geometry the angle sum of a triangle is lt 180 ln Euclidean geometry the angle sum of a triangle is 180 ln Elliptic geometry the angle sum of a triangle is gt 180 Homework 27 12

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