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# FOUNDATIONS OF GEOMETRY MATH 531

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This 5 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 531 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/229541/math-531-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

Axioms and Basic Results in Plane Geometry Unde ned Terms Point line incidence between congruent Incidence Axioms IA 1 For any pair of distinct points P and Q there is a unique line through P and Q IA 2 Any line has at least two points on it IA 3 There are three points A B 0 not all on the same line etweenness Axioms BA 1 If A gtk B gtk C then A B and C are three distinct points all lying on the same line and O gtk B gtk A gt BA 2 Given any two points B and D there exist points A C and E on BD so that ABD BODandBDE BA 3 If A B C are distinct points on the same line then exactly one of them is between the other two De nition If A and B are distinct points then the segment E is the set of points X so thatXAXBorAgtkXgtkB De nition If A and B are distinct points then the ray AB is the segment E together with the set of points X so that A gtk B gtk X De nition If Z is a line and A B are points not on Z then A and B are on the same side of Z i A B or the segment AB does not meet Z They are on opposite sides of Z if A 31 B and E intersects Z BA 4 PLANE SEPARATION For any line Z and points A B and 0 not on Z i if A and B are one the same side of Z and B and C are on the same side of Z then A and C are on the same side of Z ii if A and B are on opposite sides of Z and B and C are on opposite sides of Z then A and C are on the same side of Z Congruence Axioms CA 1 If A and B are distinct points and if A is any point then for each ray 7 starting at A there is a unique point B on 7 so that B 31 A and AB g A B CA 2 If g andABg then CDgEF If g then gm Also any segment is congruent to itself CA 3 lfAgtkBgtkO A B O W and w then m CA 4 If Given any angle ltBAC where by de nition of angle AB is not opposite AC and given any ray A B there is a unique ray AC on a given side of the line A B so that ltB A O g lt1ABO CA 5 If ltA ltB and ltA lt10 then ltA B If ltA ltB then ltB ltA Also any angle is congruent to itself De nition Two triangles AABC and ADEF are congruent written AABC g ADEF i 2 W W 2 W E 2 W ltABO 2 ltDEF ltBOA 2 ltEFD and ltOBA 2 ltFDE CA 6 SAS If two sides and the included angle of a triangle are congruent respectively to two sides and the included angle of anther triangle then the two triangles are congruent Axioms of Continuity Circular Continuity Principle If a circle 7 has on point inside and one point outside of anther circle 7 then the two circles interest in two points Elementary Continuity Principle If one endpoint of a circle is inside of a circle and the other is outside then the segment interests the circle 1 2 Archimedes Axiom If E and CD are any segments then there is number n so that if segment CD is laid off 71 times on the ray AB then a point E on AB is reached where nCDgAE andAgtkBgtkE Aristotle s Axiom Given and side of an acute angle and any segment E there is a point Y on the given side of the angle such that if X is the foot of the perpendicular from Y to the other side of the angle W gt E Important Corollary to Aristotle s Axiom Let AB be any ray P an point not colinear with A and B and ltXVY any acute angle Then there exists a point B on the ray AB such that ltPRA lt ltXVY Dedekind s Axiom Suppose the set of all points of a line Z is the union X31 U 22 of two nonempty subsets such that no point of 21 is between two points of 22 and no point of 22 is between two points of 21 Then there is a unique point 0 on Z so that if P1 6 21 P2 6 22 with P1 P2 and 0 distinct then P1 gtk O gtk P2 Axiom of Parallelism Hilbert s Parallel Axiom For every line Z and every point P not on Z there is at most one line m through P and parallel to Z Basic Results About Incidence Prop 21 If Z and m are distinct lines that are not parallel then Z and m have exactly one point in common De nition A set of lines 8 is concurrent if there is a point P so that every member of Z of 8 passes through P Prop 22 There exist three distinct lines that are not concurrent Prop 23 For every line there is a point not lying on it Prop 24 For every point there is at least one line not passing through it Prop 25 For every point P there exists at least two distinct lines that pass through P Basic Results About Betweenness Prop 31 For every pair of distinct points A B i AB BA E ii AB U BAAB De nition Let Z be a line and P a point not on Z Then the half plane determined by Z and P is the set of points X that are on the same side of Z as P The half plane determined by Z and P will also be called the side of Z of P The line Z is said to bound any of its half planes Prop 32 Every line bounds exactly two half planes and these half planes have no point in common Prop33 lfAgtkBgtkOandAODthenBODandABD Corollary lfAgtkBgtkOand BODthenABDandAOD Prop 34 LINE SEPARATION PROPERTY If C s A s B and Z is the line through A B and C then for every point P lying on Z either P lies on the ray AB or the opposite ray AC Pasch s Theorem lf AABC is any triangle and Z is any line intersecting side E in a point between A and B then Z also intersects either side E or side E If C does not lie on Z then Z does not intersect both E and W Prop 35 Given Agtk Bgtk C then E ow and B is the only point common to segments E and W a a Prop 36 Given A gtk B gtk G Then B is the only point common to the rays BA and BC Also ABAO 3 De nition Given an angle ltCAB de ne a point D to be in the interior of ltCAB iff D is on the same side of AC as B and D is on the same side of AB as 0 gt Prop 37 Given an angle ltCAB and a point D lying on the line BC Then D is in the interior of ltCAB if and only if B gtk D gtk 0 Prop 38 If D is in the interior of ltCAB then a so is every point on AD except A b no point on the opposite ray to AD is in the interior of ltCAB c if C gtk A gtk E then B is in the interior of ltDAE De nition The ray fPD is between rays fTB and RV iff APB and E are not opposite rays and D is interior to angle ltCAB Crossbar Theorem lf AD is between A0 and AB then AD intersects the segment W De nition The interior of a triangle is the intersection of the interiors of its three angles A point is in the exterior of the triangle iff it is not in the interior of the triangle and does not lie on any of the sides of the triangle Prop 39 a If a ray 7 emanating from an exterior point of AABC intersects side E at a point between A and B then also 7 intersects side E or W b If a ray emanates from an interior point of AABC then if intersects one of the sides of AABC If the ray does not go through a vertex of the triangle then it only interests one of the sides of AABE Basic Results About Congruence Corollary to SAS Given AABC and segment W E E there is a unique point F on a given side of the line D E such that AABC E ADEF Prop 310 If in triangle AABC we have E E E then ltB E lt10 Prop 311 SEGMENT SUBTRACTION If A s B s O D s E s F E E W and E E W then E g W Prop 312 Given E g W then for any point B between A and 0 there is a unique point E between D and F so that E E W De nition E lt E means there is a point E between C and D with E E W Prop 313SEGMENT ORDERING a Exactly one of the following holds trichotomy E lt E EETCD or E gt W b lf lt CD and g then lt c lf lt and lt thenmlt d lf lt and ltEF thenmlt De nition If two angles ltBAD and ltCAD have a common side fPD and the two other sides AB and AC form opposite rays the angles are supplements of each other or supple mentary angles Prop 314 Supplements of congruent angles are congruent De nition An angle ltBAD is a right angle if it has a supplementary angle to which it is congruent De nition For the de nition of vertical angles see page 24 of the text Prop 315 a Vertical angles are congruent to each other b An angle congruent to a right angle is a right angle De nition Two lines are perpendicular if they intersect at right angles Prop 316 For every line Z and every point P there there is a line through P and perpen dicular to Z 4 Prop 317 ASA CRITERION FOR CONGRUENCE Given AABO and ADEF with lt1A 2 ltD ltO 2 ltF and E 2 W Then AABO 2 ADEF Prop 318 CONVERSE TO PROP 310 If in the triangle AABC we have ltB 2 ltC then E and E and AABC is isosceles Prop 319 ANGLE ADDITION Given BEG between EA and BEG ETE between ETD ETE ltOBG 2 lt1FEH and ltrGBA 2 ltHED Then ltABO 2 lt1DEF Prop 320 ANGLE SUBTRACTION Given BG between BA and BC EH between ED and EF ltOBG 2 ltFEH and ltABO 2 lt1DEF Then ltGBA 2 ltHED De nition ltABC lt ltDEF means there is a ray EG between ED and EF such that ltABO 2 ltGEF Prop 321 ORDERING OF ANGLES a Exactly one Of the following holds trichotomy ltP lt ltQ ltP 2 ltQ or ltP gt ltQ b If ltP lt ltQ and ltQ 2 ltR then ltP lt ltR c If ltP 2 ltQ and ltQ lt ltR then ltP lt ltR d If ltP lt ltQ and ltQ lt ltR then ltP lt ltR Prop 322 SSS CRITERION FOR CONGRUENCE Given AABO and ADEF HE 2 W W 2 W and E 2 W then AABC 2 ADEF Prop 323 EUCLID S FOURTH POSTULATE All right angles are congruent tO each other Some Results in Neutral Geometry Theorem 41 ALTERNATE INTERIOR ANGLE THEOREM If two lines cut by a transversal have a pair Of congruent alternate interior angles then the two lines are parallel Corollary 1 Two lines perpendicular tO the same line are parallel Hence the perpendic ular dropped from a point P not on the line Z tO Z is unique and the point at which the perpendicular intersects Z is called its foot Corollary 2 If Z is a line and P a point not on Z then there is at least one line through P and parallel tO Z Theorem 42 EXTERIOR ANGLE THEOREM An exterior angle Of a triangle is greater than either remote interior angle Proposition 41 SAA CONGRUENCE CRITERION Given triangles AABC and ADEF with E 2 W ltA 2 ltD and ltB 2 Then AABO 2 ADEF Proposition 42 Two right triangles are congruent if the hypotenuse and a leg Of one are congruent respectively tO the hypotenuse and a leg Of the other Proposition 43 MIDPOINTS Every segment has a unique midpoint Proposition 44 BISECTORS a Every angle has a unique bisector b Every segment has a unique perpendicular bisector Proposition 45 In a triangle AABC the greater angle lies Opposite the greater side and the greater side lies lies Opposite the greater angle That is E gt W iff ltC gt ltA Theorem 43 A There is a unique degree measure assigned tO each angle sO that 0 ltA is a real number between 0 and 180 90 iff ltA is a right angle gt lt15 ltIB i lt14 2 ltgtB lf AC is interior tO ltBAD then ltBAD ltDAC ltCAB For every real number x between 0 and 180 there is an angle ltA with ltA lf ltA is supplementary tO ltB then ltA ltB 180 gt lt14 lt lt18 i r lt lt13 B Given a segment OI called the unit segment then there is a unique way tO Of assigning a length tO tO each segment E sO that 7B1 is a positive real number and 1 B111i 2 ABOi ii i 0lti 11lt11 11 For every positive real number x there is a segment E with x Corollary 1 The sum of the angles of any two angles of a triangle is less than 180 Corollary 2 TRIANGLE INEQUALITY If A B and C are three non colinear points then E 1 2 1W1 Theorem 44 SACCHERI LEGENDRE The sum of the angles of a triangle is 3 180 Corollary 1 The sum of two angles of a triangles is less than or equal to the remote exterior angle De nition The quadrilateral DABCD is convex iff it has a pair of opposite sides eg AB and CD such that CD is contained in one of the half planes bounded by A B and E is contained in on of the half planes bounded by CD Corollary 2 The sum of the measures of the angles in any convex quadrilateral is at most 360 Euclid s Postulate V If two lines are intersected by a transversal in such a way that the sum of two interior angles on one side of the transversal is less than 180 then the two lines meet on that side of the transversal Theorem 45 Euclid7s fth postulate ltgt Hilbert7s parallel postulate Proposition 47 Hilbert7s parallel postulate ltgt any line that intersects one of two parallel line intersects the other one Proposition 48 Hilbert7s parallel postulate ltgt Converse to theorem 41 which is the ALTERNATE INTERIORS ANGLES THEOREM Proposition 49 Hilbert7s parallel postulate ltgt ift is transversal to both Z and m 11771 and t1l then t1m Proposition 410 Hilbert7s parallel postulate ltgt kHZ m1k and 7116 implies m n or Proposition 411 Hilbert7s parallel postulate gt the sum of every triangle is 180 De nition For any triangle AABC the defect of AABC is de ned by 6AABC 180 7 ltltlt1Agt ltlt18gt ltltgt0gt gt Theorem 46 ADDITIVITY OF THE DEFECT Let AABC be any triangle and let D be a point between A and B Then 6AABC 6AACD 6ABCD Corollary With the same hypothesis as the last theorem the angel sum of AABC is 180 iff the angle sum of both of AACD and ABCD is 180 De nition A quadrilateral DABCD is a rectangle iff all four of its angles are right angles Theorem 47 If there exists a triangle with angle sum 180 then a rectangle exists It a rectangle exists the every triangle ha angle sum equal to 180 Corollarylf there exits a triangle with positive defect then all triangles have positive defect Dgt

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