### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# FNDNS GEOMETRY I MATH 7200

UGA

GPA 3.56

### View Full Document

## 61

## 0

## Popular in Course

## Popular in Mathematics (M)

This 16 page Class Notes was uploaded by Joanne Bergnaum on Saturday September 12, 2015. The Class Notes belongs to MATH 7200 at University of Georgia taught by Staff in Fall. Since its upload, it has received 61 views. For similar materials see /class/202084/math-7200-university-of-georgia in Mathematics (M) at University of Georgia.

## Reviews for FNDNS GEOMETRY I

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/12/15

73 Appendix A1 Basic Notions We take the terms point line plane and space as unde ned We also use the concept of a set and a subset belongs to or is an element of a set In a formal axiomatic approach we use only the properties of the unde ned terms stated in the axioms and are not allowed to make conclusions based on drawings Our rst axiom regards lines planes and space as sets of points Axiom A1 Lines planes and space are sets of points Space contains all points In geometry it is common to use the terms on in passes through and lies on We say that a point is on a line rather than belongs to a line Synonymously we say that a linepasses through a given point We also say that a point is on a plane or in a plane A line lies in a plane if it is a subset of the plane ie if every point on the line is also in the plane Points that are on the same line are called collinear and if they are on the same plane they are coplanar The following axioms describe the fundamental relationships among points lines and planes The accompanied gures are only models for the relationships Axiom A2 Any two distinct points are on exactly one line Every line contains at least two points A B Figure 1 Remark It may seem that this axiom implies that all line are straight otherwise more than one line could be drawn through two points First notice that straightness has not 74 yet been de ned Second we will show in Chapter 5 an example of geometry in which lines are objects that satisfy Axiom A2 and other axioms in this section but are not straight Axiom A3 Any three noncollinear points are on exactly one plane Each plane contains at least three noncollinear points Figure 2 Axiom A4 If two points of a line are in a plane then the entire line is in the plane Figure 3 Axiom A5 In space if two planes have a point in common then the planes have an entire line in common Axiom A6 In space if there exist at least four points that are noncoplanar Remarks on the Axioms 75 Figure 4 Figure 5 o Axiom A2 is often encountered in an equivalent form T we points determine a unique line or There is one and only one line passing through two distinct points 0 Similarly an equivalent form of Axiom A3 is Three noncollinear points determine a unique plane 0 Notice the second sentence in each of the rst two axioms We know intuitively that lines and planes contain in nitely many points7 but this fact does not follow from the above axioms Additional axioms will be needed to assure in nitude of points on a line Notation for Points Lines and Planes c l5 cummezy to dangleze poznce by ceplcel leccele of me Lezm alphabet Axlom anned u y p l l x A and B ae m male 1 wlll be denoted by A 01 Lme AB Whenevez convenlenc we alphabet wlll be used Axlom A 3 mes the a plane aan be nanned duee nonwllmem poznce on the plane pol example the plane concemmg the uppel fete ABCD of the box m male 6 aan be named m am of the fullcrwmg ways plane ABC plane 3017 plane ADC and plane ABD or wmye the plane aan be named by thee adder nonwllmeez poxnce not labeled m the gule Whenever convenlenc we nnay 91w nann plane by a smgle latte ln due text only lower aaee Greek lemeze wxll be used to nanne planee Reaall the the A39 c 7 F a Figure 6 mcezyecclon ofcwo 525 mbe ee of all elennenceclnaz 812 common to both 525 We know mcmmveLv the wo bnee emnel do no mcelsecc have no poznce m common 01 mcelyecc m exaacly one pom me l5 am am Eheolem Theorem A1 If two 4mm We Woman moy mtersen m emctly m pm Proo n 64 rlw n a palm P whlch l5 on both bnee We went on ebow ma k and 2 have no adder poznce m aommon Fol the pmpoee we nee an mama proof Suppose there l5 anodnel pom Q on k and 2 ae m mule w y Axlom A 2 theta l5 a Lmque bne than and Q ante k z wlnaln oonmadaace me bypoclnene ma the bnee ale dxffelen Consequencly the exlstence of anothel mcezsecclon pom Q must be ejected n Axlom A 3 aemed l5 the thee nonwllmeez poznce detamme a plane Ale mete n b 77 N Figure 7 to determine a plane two parallel lines determine a plane However7 we need rst to de ne parallel lines DEFINITION OF PARALLEL AND SKEW LINES Two lines are parallel if they lie in the same plane and do not intersect Lines that do not intersect and are not contained in any single plane are called skew lines gt gt gt gt If 6 and m are parallel we write 6 m Notice that in Figure 4 AB DC7 AB EH7 gt gt gt gt AC FH7 but AB and DE are skew lines Later we will de ne a cube and will be able to prove these facts INVESTIGATION 1 a What is the maximum number of intersection points determined by 71 lines in the same plane b What is the maximum number of lines determined by n points Does it matter if the points are in the same plane or not Intuitive Background for Coordinate System and Distance Following GD Birkhoff s axioms see Section 117 we introduce now the concept of distance assuming the existence and properties of real numbers We start with an intuitive background which will motivate the axioms and de nitions that follow Given any line7 a coordinate system on the line can be created by choosing arbitrary point 0 on the line and corresponding to it the number 0 The point 0 is called the origin Then to the right of point 0 another point P is chosen see Figure 8 to which 78 we correspond the number 1 The segment W is called a unit segment By laying off 3 2 1 R O P I 1 Figure 8 W to the right and left of O we nd points corresponding to integers By dividing segments into an appropriate number of equal parts we can nd points that correspond to all rational numbers The reader most likely knows that any real number and not only rational numbers corresponds to some point on the line and conversely every point on the line corresponds to some real number In Chapter 3 we will show how to nd points that correspond to real numbers such as xi and Thus there is one to one correspondence between the points on a line and the real numbers Such a correspondence is called a coordinate system for a line The number corresponding to a given point P is called the coordinate of P Thus the coordinate of Q in Figure 8 is 25 If we denote the line in Figure 8 by ac we write the coordinate of Q as ccQ Thus ccQ 25 and am 73 The distance between two points can be found using the coordinates of the points For example in Figure 8 we have PQ OQ 7 OP Mg 7 ccp 25 71 15 Notice that RP can also be found by nding the difference between the coordinates of the points RP ccp 7 am 1 7 73 4 Because distance is a nonnegative number and it is cumbersome to indicate which point has the greater coordinate we use the absolute value function Thus AB lccA 7 ccBI Based on this discussion we introduce the following axiom and de nitions Axiom A7 The Ruler Postulate The points on a line can be put in one to one correspondence with the real numbers Notice that this axiom implies that every line has an in nite number of points DEFINITION OF A COORDINATE SYSTEM FOR A LINE The correspon dence in Axiom A7 is called a coordinate system for a line A line with a coordinate system is called a number line DEFINITION OF A DISTANCE BETWEEN TWO POINTS The imam between poxnce A and B denoted by AB15 me real numbez w 7 23 whale 2 and 21 ale the woldmaceyof A and B laypecmvebx m awoldmeze system In AH Name and the dmnnee between me poznce depende on the me enoeen animate system and 15 on the ponnon of the pom and eoneeponde to me numbel 1 r w e m m ongm newe wwh We say and alme 15 m mce m lengdu between two othel pom DEFINITION OF BETWEENNESS B 15 betwaen A and C If and only If A B and C 812 thneez and AB BC AC 522 hgme 7 Ln ch15 me we wane A 78 r C Figure 9 Unng the wneepc orbezweenneee rm poxnce x 15 pomble to de ne made geometnc gmey DEFINITION OF A SEGMENT The segment Econnm ofche poznce A and B and all nine poznce becween A and B length of eegnnen 71 15 me dxstence between A and B and 15 AB A 0an M M 1 every segment nee exeady one mxdpom Th1 fan can be proved usmg 0m mom an de nmone You waI be gmded m the process of ndmg a ploof m the ploblem ee Mos N rLV 80 if it seems obvious We also want you to realize that even intuitively obvious statements can be logically deduced from the axioms and de nitions EXAMPLE A1 Given two points A and B on a number line with respective coordinates am and 3 nd the coordinate wM of M the midpoint of Ab Solution Assume 33 gt am The de nition of the midpoint implies that AM MB This equation implies w M 7 am 3 7 w M 200M 00A 003 MM 14 wB2 A M B lt e gt xA xM xB Figure 10 INVESTIGATION 2 A student approaches the solution of Example A1 as follows Because AB 33 7 am and the midpoint of AB is halfway between A and B the coordinate of the midpoint should be 12wB 7 am The student realizes the answer is wrong but would like to know why and how to use her approach to obtain the correct answer How would you respond DEFINITION OF A RAY the ray E shown in Figure 11a is the union of E and the set of all points C such that B is between A and C The point A is called the endpoint of the ray The rays having a common endpoint and whose union is straight line are opposite rays In Figure 11b E and R are opposite rays DEFINITION OF AN ANGLE An angle is a union of two rays with a common endpoint The common endpoint is the vertex of the angle and the two rays are called the sides of the angle Figure 11 Ln F Xgme 12a the angle alnownlaclne unconbe and E and lcavezcem A Unng ae nbcambn the angle 15 H u E The angle m hgme 12a 15 denglaced by LBAC bl LCAB When there la no dangel of amblgm 1 la cbmmbn to name an angle by we value Thug me angle m ngme 12a can alad be denoted by LA fthe caya E and A 912 onche came lme le 11A 3 and c ale wlll 291 am hgme 12b LABS la called a straight angle Lf A B and 0 e12 mlee noncollmee pbmca men me ann of B A a C A B I I Figure 12 the chxee aegnenca AF 3239 and A la called a mangle and denoted by AABC abbwn m ngme 13 The duee aegnenca ale called me ndea of the mangle and me chxee pomts the Verb ices ch15 pom we cbnld plove aevecal nazemenca whlch e12 mmmveLv bbvmua Pb example n A 73 a 0 men 0 73 a A Alad AB BA and AABC ACBA By equal we mean exacayll equal The la m the sec chaoly aenee two sets an cqnal 1f b q n anmmely mcb two pane aebazazed by the lme and eacln la lefeued 0 a5 the lephma Tlma fem W 1w h d d a A r he r In a 0 Figure 13 de ne when mnenn by a mm m DEFINITION OF A CDNVEX SET A set 15 convex If fol every two pomci P and Q belongng to the get me encne segment mmn the 52 I b 6 Figure 14 Name the me nncenme of the mangle a5 wen a5 me uncle m hg ue 14a and n 912 convex 525 We e12 mekmg due memen on an mnunve hens Ine mneuol of a w m r loweva 1 15a and m r gme due no enmer belong to the gme PROBLEMS SDLVED AND UNSDLVED Pmpemee of convex seceheve been 83 a b Figure 15 We brie y describe here two properties of convex sets One of which has been proved and the other which has not We use terminology that should be intuitively clear but has not yet been precisely de ned Chords intersecting at 600 In the interior of any closed convex curve See Figure 18 there exists a point P and three cords through P a chord of a region is a segment connecting any two boundary points of the region with the following property The six angles formed at P are 600 each and P is the midpoint of each of the chords The proof of this statement requires more extensive study of a convex sets Figure 16 Equichordal Points A point P in the interior of a region is an equichordal point if all chords through P are of the same length For example the center of a circle is an equichordal point of the circle However7 there exists non circular regions that have an equichordal point In 1916 Fuijwara raised the question of whether there exists a plane convex region which has two equichordal points No one has been able to give a complete answer to this question In 1984 Spaltenstein described a construction of a convex region on a sphere which has two equichordal points This does not answer the question posed in 1916 as the region is on a sphere and hence not a planar Axiom A8 The PlaneSeparation Axiom Each line in a plane separates all the points of the plane which are not on the line into two nonempty sets called the half planes with the following properties i The half planes are disjoint have no points in common convex sets ii If P is in one half plane and Q is in the other the segment m intersects 6 Figure 17 Notice that neither of the half planes in Axiom A8 includes the line Thus a line divides the plane into three mutually disjoint subsets the two half planes and the line Also it follows from Axiom A8 that a half plane is determined by a line and a point not on the line Thus we can refer to the two half planes in Figure 17 as the half plane of 6 containing P and the half plane of 6 containing Using Axiom A8 it is possible to prove a theorem named after the German math ematician Moritz Pasch 1843 1930 which states that if a line intersects one side of a triangle and does not go through any of its vertices it must also intersect another side see Figure 18 In 1882 Pasch published one of the rst rigorous treatises on geometry where he stated the theorem as an axiom he did not use Axiom A8 as an axiom Pasch realized that Euclid often relied on assumptions made Visually from diagrams and contributed to lling the gaps in Euclid s reasoning We state Pasch s Axiom as a theorem and leave its proof to be explored in the problem set 35 Theorem A2 Fasch s Axiom If a line tmersects a 5142 of a mangle and does not imam any oft12 aemaea i also mmama another 5142 of the mangle P Figure 15 Unng Axiom A 8115 poanhle to prenaer de ne me mnemol of an angle and henae the lnceuol of a cuengle DEFINITION OF INTERIOR AN ANGLE U A B C me not wllmeel uen me Incenol of 43M re the lncezsecclon of me half plane of A concemmg c wxch d1 half plane of E concemmg B See Figure 19 Figure 19 Remark The above de nmon 15 vahd well de ned iii mndependen ofche choice of B and C on due ndea ofche angle I can he pzoved the chmndeed 15 the ease Using the above de nition it is possible to de ne the interior of a triangle Coming up with an appropriate de nition is left as an exercise The de nition of the interior of an angle can also be used to de ne betweenness for rays DEFINITION E BE IEYEENNESS OF RAYS E is between E and A and if and only if AB and A0 are not opposite rays and D is in the interior of ZBAC See Figure 20 Figure 20 The above de nition can be used to prove the following visually obvious theorem lts proof is left as an exercise Theorem A3 CD intersects side AB of AABC between A and B if and only if CD is A A between CA and CB See Figure 21 Figure 21 Angle Measurement Angiee ale oommoniy meemed in degeee wth e plotleLtol We Win lane Introduce Tu meanue FW 1 A 3 Figure 22 the mcmmve knowledge of the plotlactol Axiom A9 The Angie Measurement Axiom To eyeny oner LBAC corresponds o W 1w MLBAU in mine 23 MLBAU 50 end we eoy mo LBAC ie e 50 degee angle wntten 50 Axiom A1o The Angie Construction Posiuieie Let A be o my m the edge ofa na pzme For eyeiy malnmber y o lt y lt 130 there is emctly one my mm C m the halfphme smh m MLCAB V Axiom A11 The Angie Addiiion Posiuieie If D is o polmt 1nch mmm of LBAC then MLBAD MLDAC MLBAC Nome the Axiom A 11 impheechoz meenueeoi ongiee can be computed by meme non e g MLBAD MLBAU e mums Figure 23 Problem Set 11 In the following problems use the axioms de nitions and theorems of this section to prove what is required 1 Prove that a If two lines intersect they lie in exactly one plane b Two parallel lines determine a unique plane 2 Consider the unde ned terms ball player and belongs to along with the following axioms Axiom A2 There is at least one player Axiom A3 To every player belong two balls Axiom A4 Every ball belongs to three players Prove or disprove each of the following Hint You may want to model the balls by a point and players by segments

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over $500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.