1. Foundations of Geometry Lecture Notes for Math 370
California State University, Northridge Revised for Spring
2013
2. ii Foundations of Geometry: Lecture Notes to Accompany Math
370 using Venemas Geometry California State University, Northridge
Revised for Spring 2013 (last update: November 18, 2012) This
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STUDENTS: THIS IS NOT A TEXTBOOK. IF YOU ARE USING THESE NOTES,
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5. Section 1 Preface Reections Foundations of Geometry is a
content class for future high school teach- ers. In most college
math classes, the focus is on a logical development of mathematical
tools and techniques; their underlying theory, derivations and
proofs; and application to the real world or other mathematical
sub- disciplines. As a content class we will do all this, but we
will also make an (albeit limited) attempt to integrate the
material in a way that it is pertinent to your future as a an
educator. On Pedagogy. As teachers you will need to know the most
eective way to present the material to others. To be eective, you
should be clear, con- cise, and interesting. To be successful your
presentation needs to promote understanding and retention. Most of
this cannot be learned in the classroom. It requires hard work and
lots of practice. I can tell you what I nd eective but that might
not help very much. My students are more mathematically mature than
yours will be - after all, they are college mathematics majors. My
students are people who like math, want to learn more about it, and
(in the your case) are anxious to share that excitement with
others. Middle school and high school is dierent. So most of what
works for me probably wont work very well in your classrooms. As
you progress towards your California teaching credential you will
take a lot of classes in pedagogy, the art or science of teaching.
Some of these will focus on pedagogy in mathematics and others will
not. A few of you 1
6. 2 SECTION 1. PREFACE are enrolled in integrated programs
that teach content simultaneously with pedagogy (but in dierent
classes), or already have teaching experience and want a refresher
in the content area. To these students what I have to say here is
probably nothing new. Integrating Content with Pedagogy. Our focus
is on mathematical content. But can content be separated from
pedagogy? Should it be? This is not an easy question. Some would
say that as mathematicians we should leave sociology to the
sociologists. This has a grain of truth to it - we can best teach
what we know best - and instructors might do more harm than good by
straying outside their own specialized disciplines. It seems nearly
everybody has an opinion on the matter. But to leave pedagogy out
of the equation completely would be you, as a future teacher, a
disservice. This is not a class in pedagogy. I am not going to tell
you how to teach. I couldnt even if I wanted to. But what I want
you to do as you progress through the semester is to think about
how you can communicate your knowledge to others. Start with your
colleagues in this class. If you really understand something you
should be able to explain it to people with back- grounds similar
to yours. Ask for their criticism and listen to what they say. Take
their suggestions into account the next time you work together. Of
course being able to explain your proof of the Pythagorean theorem
to the student sitting next to you does not mean youll be able to
explain it to your ninth grade students. But it will get you
thinking about explaining things in your own words. Listen to what
you say. Explain things the way you would want them explained to
you. Think about what questions you might have, and answer them to
yourself. As a teacher in training, you should continually be
asking yourself how am I ever going to teach this? whenever you are
presented with new mathematical content. Yes, you should certainly
master the material. Yes, you should learn the big picture - how
does this new content mesh with what you already know about math?
Where else can we go from here? What other theorems, results, and
methods can we obtain? How can this content be applied to other
applications outside pure math? Topics and Presentation. This class
will give you the big picture. Youll get down and dirty in other
classes that are more focused on spe- cic grade level requirements.
How to present this big picture has a long and well storied
history. You could say it all started with Euclid but all he really
did was write CC BY-NC-ND 3.0. Revised: 18 Nov 2012
7. SECTION 1. PREFACE 3 down in a formal manner the knowledge
of his day. He started by laying out lists of assumptions and basic
axioms, and then asking what could be derived, using the logical
methods of his day, from those assumptions. The result has come to
be known as Euclidean geometry. Euclid boiled the basics down to ve
axioms (after writing a whole mess of somewhat confusing
denitions). And followed it with a zillion pages of theorems. In
the past century or so other mathematicians have tried to
re-formulated Euclids axioms so that they are easier to learn.
These other presentations, starting with Hilbert and continuing up
to the common core standards used today, are all completely
equivalent, in the sense that the same mass of mathematical
knowledge is derivable from them. This process of re- dening Euclid
to make things easier to teach and learn has not been without
controversy. On Standards. The Common Core State Standards
Initiative has now been adopted by 45 states, the District of
Columbia, Guam, American Samoa, and the U.S. Virgin islands.1 These
standards give teachers (or schools, or boards of education,
depending upon where you end up teaching) a framework within which
to begin your presentation. These recommenda- tions serve as a
resource for educators, and help provide a guide for the de-
velopment of curriculum, instruction, and assessment. Since the the
choice of textbook will, in all likelihood, be determined by
factors outside your control, the formal method of presentation may
also be pre-determined. The common core standards adopted in
California for a high school geome- try class give a list of 48
specic topics that should be covered divided into categories that
include congruence, similarity, right triangles, circles, arc
lengths, measurement and dimension (among others). As you go
through this course I want you to think about how what we are
studying ts into this material. Ask yourself questions like this:
how can I put this together into a coherent, logical sequence? How
can I make it interesting and fun, challenging and not boring? How
can I promote comprehension? How can I make my students more
autonomous? How can I communicate high ex- pectations to all my
students without bias? How can I treat all my students equally? How
can I assess my students retention and understanding? How can
technology help? What sort of examples would make this easier to
un- derstand? What sort of activities would promote retention? What
should go into my lesson plan? These are not easy question and
there are no easy answers. Thinking about them now, while you are
learning geometry for the rst time (or perhaps 1Holdouts are
Alaska, Texas, Nebraska, Minnesota, Virginia, and Puerto Rico.
Revised: 18 Nov 2012 CC BY-NC-ND 3.0.
8. 4 SECTION 1. PREFACE re-learning it from a dierent
perspective) should help make you better teachers in the end. There
is a lot of work to be done, and a long and bumpy road ahead of
you.However, the more traditional approach in our state is to learn
the content material rst, then study pedagogy. Spring 2012
Reections The notes have been only marginally modied this semester,
primarily to reect an easier to read page size2 . I have also run
things through the spell-checker and xed probably half of the
spelling er- rors. Unfortunately the grammar and style is still
atrocious. I did have fun discovering such interest- ing spelling
corrections that were suggested such as circumciser for
cirumcenter; orthodontic for ortho- center; verticals for vertices;
Ne- braska for Bhaksara; saccharine for Saccheri; Stouer for Ploue;
ge- ography for geogebra; conscate for Chudnuvsky; and Yardmaster
Kandahar for Yasumasa Kanda- har. The Mohr-Moscheroni Theo- rem
became the More Maraschinos Theorem; Clairuts Axiom the Clairol
Axiom; Boylais Theorem the Bonsai Theorem; and our venerable
textbook by Venema, was reduced to a piti- ful enema. I also
discovered that non-mathematicians spell pointwise as two words or
with a dash, as in point-wise, and that I cant spell isosceles
worth beans. I guess, like the scarecrow, I deserve my Th.D.
(Doctorate of Thinkology), for that. Spring 2013. Ive xed all of
the bugs that were discovered last spring, added some stu on the
common core, and shortened the original preface, though I think its
still too wordy. 2That is, for me, with low vision, standing in
front of the class, lecturing. CC BY-NC-ND 3.0. Revised: 18 Nov
2012
9. Section 2 NCTM Math Recommendations The National Council of
Teachers of Mathematics (http://www.nctm.org) is the worlds largest
mathematics education organization, with almost 100,000 members.
They have pub- lished several highly inuential and at times,
controversial standards over the past several years. The rst set of
curriculum standards was released in 1989; professional standards
for teach- ers in 1991; and assessment standards in 1995. A more
focused set of content standards was released in their 2000 re-
port.1 Although California was one of the rst states to adopt much
of the original 1989 standards, they remain controversial, even in
our own univer- sity. The 2000 report listed four goals of its
study: 1. To set forth a comprehensive set of goals for students at
all levels (PK-12); 2. To serve as a resource for educators and
policy makers; 1Principles and Standards for School Mathematics,
NCTM, 2000, ISBN 0873534808. 5
10. 6 SECTION 2. NCTM 3. To guide in the development of
curriculum frameworks, assessments, and instruction; and 4. To
stimulate discussion at all levels on how best to help students
learn math. It recommendations included six principles for primary
and sec- ondary mathematics education and a set of content
standards for each subject area. Recommended con- tent is discussed
in the next sec- tion. The six principles of school mathematics
recommended by the NCTM were Equity, Curriculum, Teaching,
Learning, Assessment, and Technology Equity. The report challenges
the stereotype that only some students are capable of learning
mathematics, and compares it to the pervasive belief that everyone
should learn to read and write in English. This bias leads to low
expectations for many students, especially among students who live
in poverty, are non-native English speakers, have disabilities, or
are female, among other groups. The Equity Principle de- mands that
high expectations be communicated in words and deeds to all
students with instructional programs that are interesting to
students and help them see the importance of mathematical study.
Curriculum. A curriculum is more than a collection of activities:
it must be coherent, focused on important mathematics, and well
articulated across the grades. Lessons should be planned so that
fundamental ideas form an integrated whole. Teaching. Eective
mathematics teaching requires understanding what students know and
need to learn and then challenging and supporting them to learn it
well. Eective teaching requires knowing and understanding
mathematics, students as learners, and pedagogical strategies.
Teachers need to understand the big ideas in math and present them
as a coherent CC BY-NC-ND 3.0. Revised: 18 Nov 2012
11. SECTION 2. NCTM 7 and connected subject. They must know
where commen misunderstand- ings arise, and be able to present
important concepts like a fraction is part of the whole with
dierent or multiple representations of each idea or concept.
Eective teaching also requires a challenging and supportive
classroom environment. Well chosen tasks can encourage students
curios- ity. Teachers must decide which aspect of a task to
highlight, what ques- tions to ask, and how to support students
without doing the thinking for them. Finally, eective teaching
requires continually seeking improvement. You will learn much by
observing your students, listening carefully to their questions,
ideas, and explanations, and analyzing what your students are doing
and how your actions aect your students learning. Learning.
Students must learn mathematics with understanding, actively
building new knowledge from experience and prior knowledge. Factual
knowledge, procedural prociency, and conceptual understanding are
the three components to this. Memorization of facts without
understanding leaves students unable to apply these facts or
procedures to knew do- mains, while conceptual understanding
together with factual and proce- dural knowledge makes subsequent
learning easier. Autonomous learning then becomes possible, and
students can eventually take control of their own learning.
Assessment. Assessment should support the learning of important
mathematics and furnish useful in- formation to both teachers and
stu- dents. In the modern age of stan- dardized tests and numerical
met- rics for corporate quality and pro- duction this seems to have
been for- gotten. Good assessment practices provide information to
both the student and the teacher. Scoring guides or rubrics can
help teachers understand students prociencies. And poor student
performance is just as often an indication that the teacher is not
providing sucient grounding in the content being assessed. Finally,
since students have dierent learning methods dierent types of
assessments may be necessary to make an eective decision regarding
student success. Revised: 18 Nov 2012 CC BY-NC-ND 3.0.
12. 8 SECTION 2. NCTM Technology. Technology is essential in
teaching and learning mathemat- ics; it inuences the mathematics
that is taught and enhances students learning. Used properly,
technology can help students learn math. This is particularly so
where a number of programs allow students to perform the
traditional compass-and-straightedge constructions with innite
accuracy and neatness. But these constructions can do something
that construc- tions on paper cannot: students can drag points and
move lines and shapes around and see the results immediately. This
is a type of feedback never before possible and we are still
grappling with its implications and possi- bilities. The Standards.
Four basic standards are dened for all grades; at each grade level
certain expectations are to be met. The four basic geometry
standards deal with (1) geometric properties; (2) spatial
relationships; (3) transformations and symmetry; and (4) problem
solving. They are listed in more detail below, along with the
specic expectations for grades for 9 through 12. Portions of the
standards are published on-line at http:
//standards.nctm.org/document/ but beyond this list specic details
are lacking from the report; the entire geometry standard is only
10 pages.2 Details are given in the box on the following page.
Figure 2.1: Part of an example analyzing properties of two- and
three- dimensional geometric shapes to develop mathematical
arguments about geometric relationships from the NCTM Standards for
high school geome- try. [gure from NCTM Standards (2000), Chapter
7] 2For whatever reason NCTM has chosen not to make these details
open-source but instead require membership ($53/year) or document
purchase ($54.95) for complete access. Thus many teachers never see
this document. There is a copy in the CSUN library, though. CC
BY-NC-ND 3.0. Revised: 18 Nov 2012
13. SECTION 2. NCTM 9 California Framework 1. Analyze
characteristics and properties of two- and three-dimensional
geometric shapes and develop mathematical arguments about geometric
relationships to: (a) analyze properties and determine attributes
of two- and three-dimensional ob- jects; (b) explore relationships
(including congruence and similarity) among classes of two- and
three-dimensional geometric objects (gure 2.1), make and test
conjectures about them, and solve problems involving them; (c)
establish the validity of geometric conjectures using deduction,
prove theorems, and critique arguments made by others; and to (d)
use trigonometric relationships to determine lengths and angle
measures. 2. Specify locations and describe spatial relationships
using coordinate geometry and other representational systems: (a)
use Cartesian coordinates and other coordinate systems, such as
navigational, polar, or spherical systems, to analyze geometric
situations; (b) investigate conjectures and solve problems
involving two- and three-dimensional objects represented with
Cartesian coordinates. 3. Apply transformations and use symmetry to
analyze mathematical situations: (a) understand and represent
translations, reections, rotations, and dilations of objects in the
plane by using sketches, coordinates, vectors, function notation,
and matrices; (b) use various representations to help understand
the eects of simple transforma- tions and their compositions. 4.
Use visualization, spatial reasoning, and geometric modelling to
solve problems: (a) draw and construct representations of two- and
three-dimensional geometric objects using a variety of tools; (b)
visualize three-dimensional objects and spaces from dierent
perspectives and analyze their cross sections; (c) use vertex-edge
graphs to model and solve problems; (d) use geometric models to
gain insights into, and answer questions in, other areas of
mathematics; (e) use geometric ideas to solve problems in, and gain
insights into, other disciplines and other areas of interest such
as art and architecture. Revised: 18 Nov 2012 CC BY-NC-ND 3.0.
14. 10 SECTION 2. NCTM CC BY-NC-ND 3.0. Revised: 18 Nov
2012
15. Section 3 California Framework and Standards In 1997 the
California State Board of Education adopted Mathematics Content
Standards for Public Schools. These standards list specic content
material that should be covered at each level of public education
in Mathemat- ics. This material has since been su- perseded by the
Common Core Stan- dards, which were adopted by Califor- nia in
2010. by meeting the goals of standards-based mathematics, students
will achieve greater prociency in the practical uses of mathematics
in every- day life, such as balancing a check book, purchasing a
car, and understanding the daily news. Furthermore, when students
delve deeply into mathemat- ics, they gain not only conceptual
understanding of mathematical principles but also knowledge of and
experience with pure reasoning. One of the most important goals of
mathematics is to teach students logical reasoning1 . 1If you dont
know how to add fractions, you dont know how to add.[Attributed to
11
16. 12 SECTION 3. CA STANDARD In 2005 California also adopted a
Mathematics Framework for California Public Schools. The purpose of
this framework was to provide for in- structional programs and
strategies, instructional materials, professional development, and
assessments that are aligned with the standards with the intent of
providing a context for continuing a coordinated eort to en- able
all California students to achieve rigorous, high levels of
mathematics prociency. These are largely derived from the NCTM
standards, and are summarized in the boxes on this and the
following pages. An important part of the framework is balance. A
balanced program should give the student (a) prociency in basic
skills; (b) conceptual understand- ing; and (c) problem solving
ability. 1. Become procient in basic computational and procedural
skills. These are the basic skills that all students should learn
to use routinely and automatically. They should be practiced
suciently and used frequently enough to commit them to memory and
ensure that these skills are retained and maintained over the
years. 2. Develop conceptual understanding. Students who do not
have a deep understanding of mathematics suspect that it is just a
jumble of unrelated procedures and incomprehensible formulas. In
seeing the larger picture and in understanding the underlying
concepts, students are in a stronger position to apply their
knowledge to new situations and problems and to recognize when they
have made procedural errors. 3. Become adept at problem solving.
Problem solving in mathematics is a goal-related activity that
involves applying skills, understandings, and experiences to
resolve new, challenging, or perplexing mathematical situa- tions.
Problem solving involves a sequence of activities directed toward a
specic mathematical goal, such as solving a word problem, a task
that of- ten involves the use of a series of mathematical
procedures and a conceptual representation of the problem to be
solved. All three components are important; none is to be neglected
or under- emphasized. Balance, however, does not imply allocating
set amounts of time for each of the three components. professor
Barry Simon, Caltech, in a freshman mathematics class.] CC BY-NC-ND
3.0. Revised: 18 Nov 2012
17. SECTION 3. CA STANDARD 13 Goals for Teachers (2005 CA.
Framework) 1. Increase teachers knowledge of mathematics content
through professional develop- ment focusing on standards-based
mathematics. 2. Provide an instructional program that preserves the
balance of computational and procedural skills, conceptual
understanding, and problem solving. 3. Assess student progress
frequently toward the achievement of the mathematics stan- dards
and adjust instruction accordingly. 4. Provide the learning in each
instructional year that lays the necessary groundwork for success
in subsequent grades or subsequent mathematics courses. 5. Create
and maintain a classroom environment that fosters a genuine
understanding and condence in all students that through hard work
and sustained eort, they can achieve or exceed the mathematics
standards. 6. Oer all students a challenging learning experience
that will help to maximize their individual achievement and provide
opportunities for students to exceed the standards. 7. Oer
alternative instructional suggestions and strategies that address
the specic needs of Californias diverse student population. 8.
Identify the most successful and ecient approaches within a
particular classroom so that learning is maximized. Goals for
Students (2005 CA. Framework) 1. Develop uency in basic
computational and procedural skills, an understanding of
mathematical concepts, and the ability to use mathematical
reasoning to solve math- ematical problems, including recognizing
and solving routine problems readily and nding ways to reach a
solution or goal when no routine path is apparent. 2. Communicate
precisely about quantities, logical relationships, and unknown
values through the use of signs, symbols, models, graphs, and
mathematical terms. 3. Develop logical thinking in order to analyze
evidence and build arguments to support or refute hypotheses. 4.
Make connections among mathematical ideas and between mathematics
and other disciplines. 5. Apply mathematics to everyday life and
develop an interest in pursuing advanced studies in mathematics and
in a wide array of mathematically related career choices. 6.
Develop an appreciation for the beauty and power of mathematics.
Revised: 18 Nov 2012 CC BY-NC-ND 3.0.
18. 14 SECTION 3. CA STANDARD The California Standards (1997)
The California Board of Education sets forth 22 specic requirements
for geometry in secondary education.2 These specics are listed on
pages 42- 43 of Mathematics Content Standards for California Public
Schools. The Mathematics Framework for California Public Schools
expands on these 22 requirements with specic examples (pages 85-89
of the Framework) and discuss specic considerations for high school
geometry on pages 184-189 of the Framework. California Framework 1.
Introduce students to the basic nature of logical reasoning. 2. Use
inductive reasoning and geometric constructions to build up a
breadth of knowledge from from a few basic axioms. 3. Become
procient in proofs and learn the basic principles of plane
geometry. 4. Study the basic properties of triangles,
quadrilaterals, circles, and parallel lines. 5. Use the system
built up to prove the Pythagorean theorem and from there develop
and understanding of areas of dierently shaped objects. 6. Use
coordinates and shapes as a natural jumping o into trigonometry. 7.
Show that the proofs of geometry can be repeated analytically using
trigonom- etry. 8. Develop the connection between geometry and
algebra, introducing the con- cepts of analytic geometry. According
to the framework document, the main purpose of the geome- try
curriculum is to develop geometric skills and concepts and the
ability to construct formal logical arguments and proofs in a
geometric setting. The geometry skills and concepts developed in
this discipline are useful to all students. Aside from learning
these skills and concepts, students will develop their ability to
construct formal, logical arguments and proofs in geometric
settings and problems. The curriculum is weighed towards plane
Euclidean geometry but allows (and encourages) some use on coor-
dinate systems and transformations. The considerations section
(pages 184-189 of the Framework) walk us through the standards and
give us a perspective on how a geometry course might be structured.
2This material has been largely superseded by the adoption of the
common core standards by California in 2010. CC BY-NC-ND 3.0.
Revised: 18 Nov 2012
19. SECTION 3. CA STANDARD 15 California Geometry Content
Standards (1 of 2) 1. Students demonstrate understanding by
identifying and giving examples of undened terms, axioms, theorems,
and inductive and deductive reasoning. 2. Students write geometric
proofs, including proofs by contradiction. 3. Students construct
and judge the validity of a logical argument and give coun-
terexamples to disprove a statement. 4. Students prove basic
theorems involving congruence and similarity. 5. Students prove
that triangles are congruent or similar, and they are able to use
the concept of corresponding parts of congruent triangles. 6.
Students know and are able to use the triangle inequality theorem.
7. Students prove and use theorems involving the properties of
parallel lines cut by a transversal, the properties of
quadrilaterals, and the properties of circles. 8. Students know,
derive, and solve problems involving the perimeter, circum-
ference, area, volume, lateral area, and surface area of common
geometric gures. 9. Students compute the volumes and surface areas
of prisms, pyramids, cylinders, cones, and spheres; and students
commit to memory the formulas for prisms, pyramids, and cylinders.
10. Students compute areas of polygons, including rectangles,
scalene triangles, equilateral triangles, rhombi, parallelograms,
and trapezoids. 11. Students determine how changes in dimensions
aect the perimeter, area, and volume of common geometric gures and
solids. 12. Students nd and use measures of sides and of interior
and exterior angles of triangles and polygons to classify gures and
solve problems. 13. Students prove relationships between angles in
polygons by using properties of complementary, supplementary,
vertical, and exterior angles. 14. Students prove the Pythagorean
theorem. 15. Students use the Pythagorean theorem to determine
distance and nd missing lengths of sides of right triangles. 16.
Students perform basic constructions with a straight edge and
compass, such as angle bisectors, perpendicular bisectors, and the
line parallel to a given line through a point o the line. 17.
Students prove theorems by using coordinate geometry, including the
midpoint of a line segment, the distance formula, and various forms
of equations of lines and circles. 18. Students know the denitions
of the basic trigonometric functions dened by the angles of a right
triangle. They also know and are able to use elementary
relationships between them. For example, tan(x) = sin(x)/cos(x),
(sin(x))2 + (cos(x))2 = 1. Revised: 18 Nov 2012 CC BY-NC-ND
3.0.
20. 16 SECTION 3. CA STANDARD California Geometry Content
Standards (Continued) 19. Students use trigonometric functions to
solve for an unknown length of a side of a right triangle, given an
angle and a length of a side. 20. Students know and are able to use
angle and side relationships in problems with special right
triangles, such as 30 , 60 , and 90 triangles and 45 , 45 , and 90
triangles. 21. Students prove and solve problems regarding
relationships among chords, se- cants, tangents, inscribed angles,
and inscribed and circumscribed polygons of circles. 22. Students
know the eect of rigid motions on gures in the coordinate The
Dragon of Proof CC BY-NC-ND 3.0. Revised: 18 Nov 2012
21. Section 4 Common Core The Common Core Standards Initiative
has been an attempt to unify the variously confusing and conicting
state curricula and to bring them into alignment with one another.
It is coordinated by the National Gover- nors Association for Best
Practices (NGA) and the Council of Chief State Schools Ocers
(CCSSO). These standards1 attempt to dened the knowl- edge and
skills that students should acquire during their K-12 education, in
order to survive basic college curricula once they graduate. Common
Core Standards: Goals 1. Are aligned with college and work
expectations; 2. Are clear, understandable and consistent; 3.
Include rigorous content and application of knowledge through
high-order skills; 4. Build upon strengths and lessons of current
state standards; 5. Are informed by other top performing countries,
so that all students are prepared to succeed in our global economy
and society; and 6. Are evidence-based. The mathematics standards
are progressive, with each level keyed to the next, so that upon
successful completion at any given level, the student is prepared
for the subsequent level. They provide for a strong foundation in
numbers and basic operations (addition, subtraction,
multiplication, divi- sion) and fractions at the elementary school
levels; geometry, algebra, and probability and statistics in middle
school; and and am emphasis on math- ematical modeling (applying
math to practical problems) in high school. 1see
http://www.corestandards.org/about-the-standards 17
22. 18 SECTION 4. COMMON CORE At all levels the standards
stress both procedural skill and conceptual un- derstanding. The
state of California has rewritten its content standards in terms of
the common core.2 California Common Core Standards for High School
Geometry Congruence Experiment with transformations in the plane
Understand congruence in terms of rigid motions Prove geometric
theorems Make geometric constructions Similarity, Right Triangles,
and Trigonometry Understand similarity in terms of similarity
transformations Prove theorems involving similarity Dene
trigonometric ratios and solve problems involving right triangles
Apply trigonometry to general triangles Circles Understand and
apply theorems about circles Find arc lengths and areas of sectors
of circles Expressing Geometric Properties with Equations Translate
between the geometric description and the equation for a conic
section Use coordinates to prove simple geometric theorems
algebraically Geometric Measurement and Dimension Explain volume
formulas and use them to solve problems Visualize relationships
between two-dimensional and three-dimensional objects Modeling with
Geometry Apply geometric concepts in modeling situations What the
standards NCTM, California, or Common Core do NOT specify is how to
develop these concepts. Whether one should follow an axiomatic
Euclidean development, or the logical framework established by
Hilbert, or start with the smaller sets of axioms developed by
groups such as the School Mathematics Study Group (SMSG), the
University of Chicago School Mathematics Group (UCSMG) is left up
to the individual school (teacher, district, etc). As a teacher
learning the advanced mathematical content behind these standards,
it is important for you to understand each of these perspectives so
that you will understand where the framework that is actually in
use in your eventual school placement ts into the big picture. 2See
http://www.cde.ca.gov/be/st/ss/. More detail is given in the
standard; the box given here is just from the summary page. CC
BY-NC-ND 3.0. Revised: 18 Nov 2012
23. Section 5 Logic and Proof in Mathematics This section is
intentionally concise as it should be a review of Math 320. The
language of mathematics is formal. Statements can be written down
in a form that separates their content from their meaning in order
to establish consistency and validity. We start with a set of
undened terms that are accepted as given without further
explanation, such as point, line, plane. Usually these terms can be
dened in some further reduced terms but, like a dictionary, we will
eventually run into circular denitions if we attempt to continue to
rene the denitions, or we end up with a statement that doesnt
really make much sense: A point is that of which there is no part.
[Euclid, denition 1] Does this really clarify what a point is? Thus
it is best to choose our undened terms as something that is
more-or-less agreed upon. Following the undened terms, we can dene
additional objects in terms of the undened. We then need to state
our assumptions. These are called postulates or axioms; in Euclids
system there are ve. Next, we have a system of rules for obtaining
new true statements from our postulates. This is our logical
system. We will sometimes dene new symbols as shorthands to
represent parts of our logical system. The true statements are
called theorems, and the sequence of steps that justies the
validity of the theorem is called a proof. We will write all of
19
24. 20 SECTION 5. LOGIC AND PROOF our theorems in the following
manner: If [hypothesis] then [conclusion] We can give our
statements names, like A and B, in which case we write: A B which
we read as If A then B or A implies B. For a theorem to be accepted
as a true, it must have a proof. A is a list of statements that
justies a theorem. Each step must be justied (or explained) by one
of the following methods: By hypothesis ... (assume that ...) By
axiom X ... (or theorem, denition, postulate, ...) By step Y ...
(an earlier step in the proof) By a rule of symbolic logic We will
discuss some of the rules of symbolic (formal) logic shortly. There
are two special types of theorems: a lemma, and a corollary. Logi-
cally there is no dierence between a theorem, a lemma, and a
corollary. A lemma is a theorem which is not really interesting
(according to the author) in itself, or is a result that is not
pertinent to the subject at hand, but is only stated only because
it makes the proof of some other theorem more interesting. A
corollary is a theorem that follows almost immediately as a result
of another theorem with very little proof. A statement A in our
logical system will only be allowed to have two values: True and
False (we may denote these values by T and F). Just writing a
statement does not make it true: If ABC is any triangle then it is
equilateral would have a truth value of False. The value of any
implication (A B) is given by the following truth table: A B A B
true true true true false false false true true false false true
The negation operation ( A) turns true to false and false to true.
We use negation to prove theorems according to method of RAA
(Reductio ad CC BY-NC-ND 3.0. Revised: 18 Nov 2012
25. SECTION 5. LOGIC AND PROOF 21 absurdum): To prove that A B,
assume B and deduce something that is not true. We dene the
operations of and (AB) and or (AB) as meaning follows: A B A B A B
T T T T T F F T F T F T F F F F We can remember this truth table by
the following rule: AB means either A is true or B is true, or both
are true; and A B means both A and B are true. The Law of Excluded
Middles says that for any statement A, either A is true, or A is
true: P, P P The converse of the theorem A B is B A. The converse
is a com- pletely dierent statement, and may or may not be true. If
both a theorem and its converse are true, we call the theorem a
logical equivalence. We write this as A B or A i B is read as A if
and only if B and means (A B) (B A) The contrapositive of a theorem
A B is B A. The contra- positive is equivalent to the original
theorem. This is demonstrated by the following truth table. A B A B
A B A B T T T F F T T F F F T F F T T T F T F F T T T T The
universal quantier x is read as for all x. The statement (x)(S(x))
means for all x, the statement S(x) is true. The existential
quantier y is read as there exists y. Revised: 18 Nov 2012 CC
BY-NC-ND 3.0.
26. 22 SECTION 5. LOGIC AND PROOF Rene Descartes explaining
math to Queen Christina of Sweden (Pierre Louis Dumesnil,
1698-1781); copy by Nils Forsberg (1884). We can then derive the
following additional rules of logic using truth tables. ( A) A
(5.1) (A B) A B (5.2) A B A B (5.3) (A B) ( A) ( B) (5.4) [xF(x)] x
F(x) (5.5) [xF(x)] x F(x) (5.6) The following rule is known as the
rule of detachment or modus ponens, [A (A B)] B (5.7) In other
words, if A is true, and we know that A B, then we also know CC
BY-NC-ND 3.0. Revised: 18 Nov 2012
27. SECTION 5. LOGIC AND PROOF 23 that B is true. We also have
the following rules of deduction: [(A B) (B C)] [A C] (5.8) [A B] A
(5.9) For the following example, recall that a rational number is
any number x that can be expressed as the ratio of two integers,
e.g., x = p/q, where both p and q are integers. An irrational
number is a number that is not rational. Example 5.1 Prove that 2
is irrational. Proof. 1. Assume that 2 is rational. (RAA
hypothesis) 2. There exists integers p and q such that 2 = p/q
(Denition of a rational number) 3. Assume that p and q have no
common factors (from our knowledge of numbers we know that we can
cancel out all common factors). 4. Therefore at least one of p and
q is odd. (otherwise there would be a common factor of 2, which
contradicts step 3 5. Since p2 = 2q2 , p is divisible by 2, hence
it is even. 6. Write p = 2s where s is an integer (denition of
even). 7. Then 4s2 = p2 = 2q2 q2 = 2s2 8. Then q2 is divisible by
2, hence it is even (denition of even). 9. Hence q is even because
the square of any odd number must be odd. 10. Since p and q are
even this contradicts step 4. Hence our RAA as- sumption is false.
We numbered the steps in this proof for clarity. In general we dont
do this, because it is somewhat tedious and takes up a lot of
space, and instead usually write them as a paragraph. However,
until you are comfortable writing proofs, you should write them out
in a step-by-step manner. Revised: 18 Nov 2012 CC BY-NC-ND
3.0.
28. 24 SECTION 5. LOGIC AND PROOF I Geometry! CC BY-NC-ND 3.0.
Revised: 18 Nov 2012
29. Section 6 The Real Numbers We will take the terms set and
element as undened terms. We write a set as the list of elements
surrounded by curly brackets: {A, B, C, ...} or by a rule {x|S(x)}
where S(x) is some rule such as x is even. We use x S to represent
x is an element of the set S. We denote the natural numbers by N =
{1, 2, 3, . . . }, the integers by Z = {. . . , 3, 2, 1, 0, 1, 2,
3, . . . }, and the positive integers by Z+ = {0, 1, 2, ..} The
union of two sets S and T is given by S T = {x|x S x T} 0This
section is intended primarily as a review and hence is necessarily
concise. 25
30. 26 SECTION 6. REAL NUMBERS The intersection of two sets S
and T is given by S T = {x|x S x T} We will use the notation A B to
indicate set dierence, which we read a A minus B A B = {x|(x A) (x
B)} Venema (and some other texts) use the notation A B for this
set. The symbol represents the empty set. The symbol Q represents
the set of all rational numbers. A rational number r is a quotient
of two integers p and q, where r = p/q The symbol R represents the
set of all real numbers. We will not give a denition of real
numbers, but example 5.1 shows that there are numbers that are not
rational. Any real number that cannot be expressed as a rational
number is called an irrational number. We will see later that there
is a one-to-one correspondence between the points on a line and the
real numbers, so in a sense, the real numbers give us anything we
can measure. Axiom 6.1 (Trichotomy of the Real Numbers) Let x, y R.
Then exactly one of the following is true: x < y, x = y, or x
> y Axiom 6.2 (Density) Let x < y R. Then both of the
following are true: 1. There exists a rational number q such that x
< q < y 2. There exists an irrational number z such that x
< z < y Corollary 6.3 There is an irrational number between
any two rational numbers. Corollary 6.4 There is a rational number
between any two irrational num- bers. Theorem 6.5 (Comparison
Theorem) Suppose that x, y R satisfy 1. For every rational number q
< x, q < y 2. For every rational number q < y, q < x
then x = y. CC BY-NC-ND 3.0. Revised: 18 Nov 2012
31. SECTION 6. REAL NUMBERS 27 Denition 6.6 (Upper Bound) A
number M is called an upper bound for a set A if x A, x M. Denition
6.7 (Least Upper Bound) A number m is called a least upper bound
for a set A if for all upper bounds M of A, m M, and we write m =
lubA. Axiom 6.8 (Least Upper Bound Axiom) Every bounded non-empty
subset of the real numbers has a least upper bound. Babylonian clay
tablet YBC 7289 (c 1800-1600 BC) showing 2 1 + 21 60 + 51 602 + 10
603 (gure Bill Casselman http://www.math.ubc.ca/
~cass/Euclid/ybc/ybc.html.) The following property expresses the
notion that you can ll up a bucket with spoonfuls of water. We will
accept it as an axiom although it fact it can be derived from the
Least Upper Bound Axiom. Axiom 6.9 (Archimedian Property) If M >
0, > 0 are both real numbers than there exists a postive integer
n such that n > M. Denition 6.10 (Function) A function f is a
rule that assigns to each element a A an element b = f(a) B. We
call A the domain of f, and we call the subset of B to which
elements of A are mapped by f the range of f. We write f : A B.
Denition 6.11 A function f : A B is one-to-one (sometimes 1-1 or
(1:1)) if a1 = a2 f(a1) = f(a2) Denition 6.12 A function f : A B is
onto if (b B)(a A) such that b = f(a). Denition 6.13 A function f :
A B that is 1-1 and onto is called a one-to-one-correspondence.
Denition 6.14 A function f(x) is continuous on an interval (a, b)
if for every > 0 there exists a > 0 such that whenever |xy|
< , x, y (a, b), then |f(x) f(y)| < . Revised: 18 Nov 2012 CC
BY-NC-ND 3.0.
32. 28 SECTION 6. REAL NUMBERS The Sum of the Squares ... CC
BY-NC-ND 3.0. Revised: 18 Nov 2012
33. Section 7 Euclids Elements Euclid at the Oxford History
Museum. Here and in the following sections we will present some of
the axiomatic systems that have been used to develop geometry in
the west since Euclids time. We will not be using any of thse
specic systems in our own de- velopment but they are worth being
familiar with. Euclids Elements was written around 300 BC. It
consists of 13 Books that are a compila- tion of geometric
knowledge that had beed ac- quired over the previous several
centuries. We dont know what (if any) parts of it are orig- inal to
Euclid. The great contribution of this document is that it sets
forth a set of basic as- sumptions (ve postulates) from which all
of the remainder of the treatise is logically de- rived. Any
logical system requires one to rst dene some basic concepts which
are accepted on faith; for geometry these are thinks like points
and lines. In fact, Eculid states 23 denitions before his
postulates, and follows them with ve additional common notions,
which are statements that he expects are so obvious they can be
accepted by anyone with reason. It is worth looking at these basic
statements to get a feeling for where Euclids geometry starts, so
they are extracted below[Euclid]. 29
34. 30 SECTION 7. EUCLIDS ELEMENTS Euclids Axioms 1 Let it have
been postulated to draw a straight-line from any point to any
point. 2 And to produce a nite straight-line continuously in a
straight-line. 3 And to draw a circle with any center and radius. 4
And that all right-angles are equal to one another. 5 Euclids
Parallel postulate. And that if a straight-line falling across two
(other) straight-lines makes internal angles on the same side (of
itself whose sum is) less than two right-angles, then the two
(other) straight-lines, being produced to innity, meet on that side
(of the original straight-line) that the (sum of the internal
angles) is less than two right-angles (and do not meet on the other
side) (see gure 7.1.) Historically the 5th postulate has been
considered separate from the rst four and was followed by over 2000
years of at- tempts to derive it from them. It was not until the
19th century (by Euge- nio Beltrami in 1868) that it was shown that
postulate 5 could not be derived from the other four. In the
process, the existence of non-Euclidean geometries was proven, as
well as neutral geometry, the study of the consequences of the rst
four postulates. Euclid presumably stated his common notions to
make clear what assump- tions he was making that he thought were
obvious. Today we would prob- ably state them using algebra, but
such expressions had not been invented yet. 0Euclids line is what
we call a plane curve. 0Euclids straight-line is what we would call
a line segment. The modern concept of a line that extends innitely
in each direction was unknown to Euclid. CC BY-NC-ND 3.0. Revised:
18 Nov 2012
35. SECTION 7. EUCLIDS ELEMENTS 31 Euclids Denitions 1 A point
is that of which there is no part. 2 And a line1 is a length
without breadth. 3 And the extremities of a line are points. 4 A
straight-line2 is (any) one which lies evenly with points on
itself. 5 And a surface is that which has length and breadth only.
6 And the extremities of a surface are lines. 7 A plane surface is
(any) one which lies evenly with the straight-lines on itself. 8
And a plane angle is the inclination of the lines to one another,
when two lines in a plane meet one another, and are not lying in a
straight-line. 9 And when the lines containing the angle are
straight then the angle is called rectilinear. 10 And when a
straight-line stood upon (another) straight-line makes adjacent
angles (which are) equal to one another, each of the equal angles
is a right-angle, and the former straight-line is called a
perpendicular to that upon which it stands. 11 An obtuse angle is
one greater than a right-angle. 12 And an acute angle (is) one less
than a right-angle. 13 A boundary is that which is the extremity of
something. 14 A gure is that which is contained by some boundary or
boundaries. 15 A circle is a plane gure contained by a single line
[which is called a circumference], (such that) all of the
straight-lines radiating towards [the circumference] from one point
amongst those lying inside the gure are equal to one another. 16
And the point is called the center of the circle. 17 And a diameter
of the circle is any straight-line, being drawn through the center,
and terminated in each direction by the circumference of the
circle. (And) any such (straight-line) also cuts the circle in
half. 18 And a semi-circle is the gure contained by the diameter
and the circumference cuts o by it. And the center of the
semi-circle is the same (point) as (the center of) the circle. 19
Rectilinear gures are those (gures) contained by straight-lines:
trilateral gures being those contained by three straight-lines,
quadrilateral by four, and multilateral by more than four. 20 And
of the trilateral gures: an equilateral triangle is that having
three equal sides, an isosceles (triangle) that having only two
equal sides, and a scalene (triangle) that having three unequal
sides. 21 And further of the trilateral gures: a right-angled
triangle is that having a right- angle, an obtuse-angled (triangle)
that having an obtuse angle, and an acute-angled (triangle) that
having three acute angles. 22 And of the quadrilateral gures: a
square is that which is right-angled and equilateral, a rectangle
that which is right-angled but not equilateral, a rhombus that
which is equilateral but not right-angled, and a rhomboid that
having opposite sides and angles equal to one another which is
neither right-angled nor equilateral. And let quadrilateral gures
besides these be called trapezia. 23 Parallel lines are
straight-lines which, being in the same plane, and being produced
to innity in each direction, meet with one another in neither (of
these directions). Revised: 18 Nov 2012 CC BY-NC-ND 3.0.
36. 32 SECTION 7. EUCLIDS ELEMENTS Figure 7.1: Illustration of
Euclids Parallel postulate. The two angles and add to les than 180
degrees, hence the lines h and k meet at a point S on the same same
side of g as and . Euclids Common Notions 1 Things equal to the
same thing are also equal to one another We might write this today
as: If a = b and b = c then a = c. 2 And if equal things are added
to equal things then the wholes are equal. We might write this as:
if a = c then a + b = c + b. 3 And if equal things are subtracted
from equal things then the remainders are equal. Which we might
write this as: if a = c then a b = c b. 4 And things coinciding
with one another are equal to one another. By this Euclid meant he
could imagine picking up the picture of a triangle (or some other
object) and lay it on top of another; if they were the same then
they were considered equal (or maybe congruent). 5 And the whole
[is] greater than the part Which we might write as: if a > 0 and
b > 0 then a + b > a and a + b > b. CC BY-NC-ND 3.0.
Revised: 18 Nov 2012
37. Section 8 Hilberts Axioms David Hilbert (1862-1943) boiled
geometry down to 20 axioms which he classied into seven axioms of
connection (we now use the term incidence); ve axioms of order; one
axiom of parallels; six axioms of congruence; and one axiom of
continuity. He did this because it had been discovered over the
centuries that Euclid had left out parts of his arguments and
Hilbert was attempting to ll in all the blanks. The axioms below
are taken from the lecture notes of his course in geometry given in
1898 and translated by E.J. Townsend in 1902. Hilberts system
begins with the follow- ing undened terms: point, line, plane, lie
on, between, congruent. His axioms are divided up into dierent
sub-areas of geometry: connection, order, paral- lels, congruence,
continuity, and com- pleteness. The axioms of connection de- ne
things like how points form lines and planes. The axioms of order
express the concept of betweenness of points. They are classied
into four linear ax- ioms of order and one plane axiom of or- der.
The axiom of parallels is a equiva- lent to Euclids parallel
postulate. The axioms of congruence formalize our intu- itive
notions of equivalences among line segments, angles, and triangles.
The ax- 33
38. 34 SECTION 8. HILBERTS AXIOMS iom of continuity introduces
the continuity of real numbers to geometry, and completeness tells
us that everything we can possibly know we can learn from these
axioms. Hilberts Axioms of Connection 1. Two distinct points A and
B always completely determine a straight line a. We write AB = a or
BA = a. 2. Any two distinct points of a straight line completely
determine that line; that is, if AB = a and AC = a, where B = C,
then also BC = a. 3. Three points A, B, C not situated in the same
straight line always completely deter- minea plane . We write ABC =
. 4. Any three points A, B, C of a plane , which do not lie in the
same straight line,completely determine that plane. 5. If two
points A, B of a straight line a lie in a plane , then every point
of a lies in . 6. If two planes , have a point A in common, then
they have at least a second point B in common. 7. Upon every
straight line there exist at least two points, in every plane at
least three points not lying in the same straight line, and in
space there exist at least four points not lying in a plane.
Hilberts Axioms of Order 1. If A, B, C are points of a straight
line and B lies between A and C, then B lies also between C and A.
2. If A and C are two points of a straight line, then there exists
at least one point B lying between A and C and at least one point D
so situated that C lies between A and D. 3. Of any three points
situated on a straight line, there is always one and only one which
lies between the other two. 4. Any four points A, B, C, D of a
straight line can always be so arranged that B shall lie between A
and C and also between A and D, and, furthermore, that C shalllie
between A and D and also between B and D. 5. Let A, B, C be three
points not lying in the same straight line and let a be a straight
line lying in the plane ABC and not passing through any of the
points A, B, C. Then, if the straight line a passes through a point
of the segment AB, it will also pass through either a point of the
segment BC or a point of the segment AC. Hilberts Axiom of
Parallels In a plane there can be drawn through any point A, lying
outside of a straight line a, one and only one straight line which
does not intersect the line a. This straight line is called the
parallel to a through the given point A. CC BY-NC-ND 3.0. Revised:
18 Nov 2012
39. SECTION 8. HILBERTS AXIOMS 35 Hilberts Axioms of Congruence
1. If A, B are two points on a straight line a, and if A is a point
upon the same or another straight line a1, then, upon a given side
of A on the straight line a , we can always nd one and only one
point B so that the segment AB (or BA) is congruent to the segment
A B . We indicate this relation by writing AB = A B Every segment
is congruent to itself; that is, we always have AB = AB. 2. If AB =
A B and also AB = A B , then A B = A B 3. Let AB and BC be two
segments of a straight line a which have no points in common aside
from the point B, and, furthermore, let A B and B C be two segments
of the same or of another straight line a having, likewise, no
point other than B in common. Then, if AB A B and BC = B C , we
have AC = A C . 4. Let an angle (h, k) be given in the plane and
let a straight line a be given in a plane . Suppose also that, in
the plane , a denite side of the straight line a be assigned.
Denote by h a half-ray of the straight line a emanating from a
point O of this line. Then in the plane there is one and only one
half-ray k such that the angle (h, k), or (k, h), is congruent to
the angle (h , k ) and at the same time all interior points of the
angle (h , k ) lie upon the given side of a . We express this
relation by means of the notation (h, k) = (h , k ) Every angle is
congruent to itself; that is, (h, k) = (h, k). 5. If the (h, k) =
(h , k ) and (h, k) = (h , k ) then (h , k ) = (h , k ) 6. If, in
the two triangles ABC and A B C the congruences ABV A B, AC = A C
and BAC = B A C then ABC = A B C and ACB = A C B . Hilberts Axiom
of Continuity (Archimedian Axiom) Let A1A be any point on a
straight line AB. Choose A2, A3, .. so that A1 is between A and A2,
A2 is between A1 and A3, etc, such that AA1 = A1A2 = A2A3 = Then
there always exists a certain point An such that B lies between A
and An. In more modern terms, > 0 and x > 0, then m N such
that m > x. Hilberts Axioms of Completeness To a system of
points, straight lines, and planes, it is impossible to add other
elements in such a manner that the system thus generalized shall
form a new geometry obeying all of the ve groups of axioms. In
other words, the elements of geometry form a system which is not
susceptible of extension, if we regard the ve groups of axioms as
valid. Revised: 18 Nov 2012 CC BY-NC-ND 3.0.
40. 36 SECTION 8. HILBERTS AXIOMS CC BY-NC-ND 3.0. Revised: 18
Nov 2012
41. Section 9 Birkho/MacLane Axioms Birkho (left); MacLane
(right) Geroge Birkho (1884-1944) is best known for his works on
dierential equa- tions, taught at Univ. of Wisconsin, Princeton,
and Harvard. In 1932 he pro- posed a very compact set of axioms
which allow you to use a ruler (a straight edge with marks on it)
and a protractor. His purpose was to make geometry more un-
derstandable to high school students.1 Saunders MacLane (1909-2005)
was a friend (and professional collaborator) of George Birkhos son
Garrett, and worked primarily at Harvard (where he met the Birkhos)
and the Univ. of Chicago. 2 In 1959 [MacLane, 1959] proposed an
extension of Birkhos Axioms that included a distance measure
thereby making the system some- what more intuitive than Hilberts.
MacLane introduced the concept of distance metrics into the axioms,
and added an axiom of continuity. 1The system was published in the
paper A Set of Postulate for Plane Geometry based on a Scale and
Protractor, in Annals of Mathematics, 33:329-345 (1932). 2he photo
is by Konrad Jacobs (CCASA license) and from Wikimedia). 37
42. 38 SECTION 9. BIRKHOFF/MACLANE AXIOMS Birkhos Axioms
Undened Terms: point, line, distance, angle. 1. Axiom of Line
Measure. The points A, B, .. of any line l can be put into (1, 1)
correspondence with the real numbers x so that |xB xA| = d(A, B)
for all points A, B. Here d(A, B) denotes the distance between the
points A and B. In other words, you are allowed to use a rule to
measure the length of a line. 2. The point-line postulate. One and
only one straight line l contains two given points P, Q (P = Q). 3.
The Axiom of angle measure. The half lines l, m, .. through any
point O can be put into (1,1) correspondence with teh real numbers
a mod 2 so that, if A = 0 and B = O are points of l and m
respectively, the dierence am al mod 2 is A0B. 4. The Postulate of
triangle similarity. If in two triangles, ABC and A B C , of for
some constant k > 0, d(A , B ) = kd(A, B), d(A , C ) = kd(A, C),
and also B A C = BAC then also d(B , C ) = kd(B, C), C B A = CBA, A
C B = ACB. MacLanes Axioms on Distance 1. There are at least two
points. 2. If A and B are points, d(AB) is a nonnegative number
(that gives the distance between the points). 3. For points A and
B, d(AB) = 0 if and only if A = B. 4. If A and B are points then
d(AB) = d(BA). MacLanes Axioms on Lines 1. A l ine is a set of
points containing more than one points. 2. Through two distinct
points there is one and only one l ine. 3. Three distinct points on
a line if and only if one of them is between the other two. 4. On
each ray from a point O and to each positive real number b there is
a point B with d(OB) = b MacLanes Axioms on Angles 1. If r and s
are rays from the same point, then rs is a real number (mod 360).
2. If r, s, t are three rays from the same point, the rs + st = rt.
3. If r is a ray from O and c is a real number, then there is a ray
s from O such that rs = c. CC BY-NC-ND 3.0. Revised: 18 Nov
2012
43. SECTION 9. BIRKHOFF/MACLANE AXIOMS 39 McLanes Axiom on
Similarity If two triangles ABC and A B C have ABC = A B C , d(AB)
= kd(A B ), and d(BC) = kd(B C ) for some postive number k then
they are similar. MacLanes Axiom of Continuity Let AOB be proper.
If D is between A adn B then 0 < AOD < AOB. Here a proper
angle is an angle x such that 0 < x < 180. The 37th View of
Mt Fuji by Katsushika Hokusai (1760-1849) never made it to wood
block. Revised: 18 Nov 2012 CC BY-NC-ND 3.0.
44. 40 SECTION 9. BIRKHOFF/MACLANE AXIOMS A Geometric Mind CC
BY-NC-ND 3.0. Revised: 18 Nov 2012
45. Section 10 The SMSG Axioms The School Mathematics Study
Group (SMSG) at Yale University was funded by the US National
Science Foundation to reform mathematics ed- ucation in the 1950s
and developed mathematical curricula that came be know as the new
math during the 1960s. A set of 22 s that were intended to make
geometry more intuitive and understandable were produced by this
group, as was a mimeographed textbook that was later used as the
basis of a geometry textbook by E.E. Moise and F.L.Downs (1964)
that is still in circulation. Some of the axioms are redundant in
the sense that they can be derived from the others. The undened
terms are point, line, plane, lie on, distance, angle measure,
area, volume, and there are 22 axioms. Axiom 1 Given any two
distinct points there is exactly one line that con- tains them.
Axiom 2 Distance Postulate. To every pair of distinct points there
cor- responds a unique positive number. This number is called the
distance between the two points. Axiom 3 Ruler Postulate. The
points of a line can be placed in a corre- spondence with the real
numbers such that: (1) To every point of the line there corresponds
exactly one real number; (2) To every real number there corresponds
exactly one point of the line. (3) The distance between two
distinct points is the absolute value of the dierence of the
corresponding real numbers. Axiom 4 Ruler Placement Postulate.
Given two points P and Q of a line, the coordinate system can be
chosen in such a way that the coordinate of 41
46. 42 SECTION 10. THE SMSG AXIOMS P is zero and the coordinate
of Q is positive. Axiom 5 Every plane contains at least three
non-collinear points, and space contains at least four non-coplanar
points. Axiom 6 If two points lie in a plane, then the line
containing these points lies in the same plane. Axiom 7 Any three
points lie in at least one plane, and any three non- collinear
points lie in exactly one plane. Axiom 8 If two planes intersect,
then that intersection is a line. Edward Grith Begle (1941-1978),
director of SMSG for ten years. Photograph by Paul Halmos, Archives
of American Mathematics, Dolph Briscoe Center for American History,
University of Texas at Austin. Axiom 9 Plane Separation Postulate.
Given a line and a plane containing it, the points of the plane
that do not lie on the line form two sets such that: (1) each of
the sets is convex; and (2) if P is in one set and Q is in the
other, then segment PQ intersects the line. Axiom 10 Space
Separation Postulate. The points of space that do not lie in a
given plane form two sets such that: (1) Each of the sets is
convex; and (2) If P is in one set and Q is in the other, then
segment PQ intersects the plane. Axiom 11 Angle Measurement
Postulate. To every angle x there corre- sponds a real number
between 0 and 180 . The real number is called the measure of the
angle and denoted by m(x). Axiom 12 Angle Construction postulate.
Let AB be a ray on the edge of the half-plane H. For every r
between 0 and 180 there is exactly one ray CC BY-NC-ND 3.0.
Revised: 18 Nov 2012
47. SECTION 10. THE SMSG AXIOMS 43 AP, with P in H such that
m(PAB) = r. Axiom 13 Angle Addition postulate. If D is a point in
the interior of BAC, then mBAC) = m(BAD) + m(DAC). Axiom 14
Supplement postulate. If two angles form a linear pair, then they
are supplementary. Axiom 15 SAS postulate. Given a one-to-one
correspondence between two triangles (or between a triangle and
itself). If two sides nd the included angle of the rst triangle are
congruent to the corresponding parts of the second triangle, then
the correspondence is a congruence. Axiom 16 Parallel postulate.
Through a given external point there is at most one line parallel
to a given line. Axiom 17 To every polygonal region there
corresponds a unique positive real number called its area. Axiom 18
If two triangles are congruent, then the triangular regions have
the same area. Axiom 19 Suppose that the region R is the union of
two regions R1 and R2. If R1 and R2 intersect at most in a nite
number of segments and points, then the area of R is the sum of the
areas of R1 and R2. Axiom 20 The area of a rectangle is the product
of the length of its and the length of its altitude. Axiom 21 The
volume of a rectangle parallelepiped is equal to the product of the
length of its altitude and the area of its base. Axiom 22
Cavalieris Principle. Given two solids and a plane. If for every
plane that intersects the solids and is parallel to the given plane
the two intersections determine regions that have the same area,
then the two solids have the same volume. Revised: 18 Nov 2012 CC
BY-NC-ND 3.0.
48. 44 SECTION 10. THE SMSG AXIOMS CC BY-NC-ND 3.0. Revised: 18
Nov 2012
49. Section 11 The UCSMP Axioms The University of Chicago
School Mathemat- ics Project was founded in 1983 with the aim of
upgrading mathematics education in el- ementary and secondary
schools throughout the United States. They have developed a set of
axioms that are in wide use today, and are also redundant in the
sense that some axioms can be proved from others. The purpose of
the redundancy was to make the learning of ge- ometry more
intuitive. These axioms used in- corporated a transformational
approach. De- tails of the projects history is given on its web
page at (http://ucsmp.uchicago.edu/ history.html). Many of todays
elementary and secondary textbooks are based on these standards,
which encompass all of mathematics, not just geometry. The only
undened terms are point, line, and plane. Point-Line-Plane Axioms
Axiom 1 Through any two points there is exactly one line. Axiom 2
Every line is a set of points that can be put into a one-to-one
correspondence with the real numbers, with any point corresponding
to zero and any other point corresponding to the number 1. Axiom 3
Given a line in a plane, there is at least one point in the plane
that is not on the line. Given a plane in space, there is at least
one point 45
50. 46 SECTION 11. THE UCSMP AXIOMS in space that is not on the
plane. Axiom 4 If two points lie in a plane, the line containing
them lies in the plane. Axiom 5 Through three non-collinear points,
there is exactly one plane. Axiom 6 If two dierent planes have a
point in common, then their inter- section is a line. Distance
Axioms Axiom 7 On a line, there is a unique distance between two
points. Axiom 8 If two points on a line have coordinates x and y
the distance between them is |x y|. Axiom 9 If point B is on the
line segment AC then AB + BC = AC, where AB, BC, AC denote the
distances between the points. Triangle Inequality Axiom 10 The sum
of the lengths of two sides of a triangle is greater than the
length of the third side. Angle Measure Axiom 11 Every angle has a
unique measure from 0 to 180 . Axiom 12 Given any ray V A and a
real number r between 0 and 180 there is a unique angle BV A in
each half-plane of V A such that BV A = r. Axiom 13 If V A and V B
are the same ray, then BV A = 0. Axiom 14 If V A and V B are
opposite rates, then BV A = 180. Axiom 15 If V C (except for the
point V ) is in the interior of angle AV B then AV C + CV B = AV B.
Corresponding Angle Axiom Axiom 16 Suppose two coplanar lines are
cut by a transversal. If two corresponding angles have the same
measure, then the lines are parallel. If the lines are parallel,
then the corresponding angles have the same measure. Reection
Axioms Axiom 17 There is a one to one correspondence between points
and their images in a reection. Axiom 18 Collinearity is preserved
by reection. Axiom 19 Betweenness is preserved by reection. CC
BY-NC-ND 3.0. Revised: 18 Nov 2012
51. SECTION 11. THE UCSMP AXIOMS 47 Axiom 20 Distance is
preserved by reection. Axiom 21 Angle measure is preserved by
reection. Axiom 22 Orientation is reversed by reection. Area Axioms
Axiom 23 Given a unit region, every polygonal region has a unique
area. Axiom 24 The area of a rectangle with dimensions l and w is
lw. Axiom 25 Congruent gures have the same area. Axiom 26 The areas
of the union of two non-overlapping regions is the sum of the areas
of the regions. Volume Axioms Axiom 27 Given a unit cube,every
solid region has a unique volume. Axiom 28 The volume of box with
dimensions l, w, and h is lwh. Axiom 29 Congruent solids have the
same volume. Axiom 30 The volume of the union of two
non-overlapping solids is the sum of their volumes. Axiom 31 Given
two solids and a plane. If for every plane which intersects the
solids and is parallel to the given plane the intersections have
equal areas, then the two solids have the same volume. Revised: 18
Nov 2012 CC BY-NC-ND 3.0.
52. 48 SECTION 11. THE UCSMP AXIOMS CC BY-NC-ND 3.0. Revised:
18 Nov 2012
53. Section 12 Venemas Axioms Venema [Venema, 2006] introduces
a set of axioms that meld together parts of the axiom systems of
Birkho, MacLane, the SMSG, and the UCSMP. We present them here for
reference, since we will be using this system in the remainder of
the class. Venemas Undened Terms The undened terms are point, line,
distance, half-plane, angle-measure, area Venemas Axioms of Neutral
Geometry Existence says that at least some points exist, and
incidents says that every pair of distinct points denes a line. 1.
Existence Postulate. The collection of all points forms a nonempty
set with more than one (i.e., at least two) points. The set of all
points in the plane is called P. 2. Incidence Postulate. Every line
is a set of points. For every pair of distinct points A, B there is
exactly one line = AB such that A, B . The ruler postulate allows
us to associate real numbers and hence mea- surements with
distances and line segments. 49
54. 50 SECTION 12. VENEMAS AXIOMS 3. Ruler Postulate. For every
pair of points P, Q there is a number PQ called the distance from P
to Q. For each line there is a one-to-one mapping f : R such that
if x = f(P) and y = f(Q) then PQ = |x y|. The Plane Separation
Postulate says that a line has two sides; it is used to dene the
concept of a half-plane. 4. Plane Separation Postulate. For every
line the points that do not lie on form two disjoint, convex
non-empty sets H1 and H2, called half-planes, bounded by such that
if P H1 and Q H2 then PQ intersects . The protractor postulate
encapsulates our common notions (to use a Euclidean term) about
angles: they can be measure, added, and ordered. 5. Protractor
Postulate. For every angle BAC there is a real number (BAC) called
the measure of BAC such that 1. 0 (BAC) < 180 2. (BAC) = 0 AB =
BC 3. Angle Construction Postulate. r R such that 0 < r <
180, and for every half-plane H bounded by AB, there exists a
unique ray AE such that E H and (BAE) = r. 4. Angle Addition
Postulate. If the ray AD is between the rays AB and AC then (BAD) +
(DAC) = (BAC) As we shall see later in the discussion, the rst ve
postulates are not sucient to ensure that our common notions about
triangle congruence will hold (for example, the following postulate
fails in taxi-cab geometry). 6. SAS (Side Angle Side Postulate) If
ABC and DEF are two triangles such that AB = DE,BC = EF, and ABC =
DEF then ABC = DEF. Parallel Postulates The combination of the rst
six postulates, when taken together, are known as neutral geometry.
They can be extended with one of three possible par- allel
postulates. It turns out that the second of these the elliptic
parallel postulate, is inconsistent with the plane separation
postulate and the con- cept of betweenness but the other two
postulates are each consistent with neutral geometry. In fact, it
can be proven that they are mutually exclusive if you accept either
one of the Hyperbolic or Euclidean parallel postulates CC BY-NC-ND
3.0. Revised: 18 Nov 2012
55. SECTION 12. VENEMAS AXIOMS 51 under neutral geometry, the
other is provably false; and if either is taken as false, the other
is provably true. Euclidean Parallel Postulate For every line and
for every external point P, there is exactly one line m such that P
lies on m and m Elliptic Parallel Postulate For every line and for
every external point P, there is no line m such that P lies on m
and m Hyperbolic Parallel Postulate For every line and for every
external point P, there are at least two lines m and n such that P
lies on both m and m and m and n . Area Postulates Neutral Area
Postulate Associated with each polygonal region R there is a
nonnegative number (R), called the area of R, such that: 1.
(Congruence) If two triangles are congruent, then their associate
re- gions have equal area; and 2. (Additivity) If R = R1 R2 and R1
and R2 do not overlap, then (R) = (R1) + (R2) Euclidean Area
Postulate (Venema 9.2.2) If R is a rectangle, the (R) = length(R)
width(R). Reection The Reection Postulate (Venema 12.6.1) For every
line there exists a transformation : P P such that: 1. If P then
(P) = P 2. If P , then P and (P) lie on opposite half planes of .
3. preserves distance, collinearity, and angle measure. Revised: 18
Nov 2012 CC BY-NC-ND 3.0.
56. 52 SECTION 12. VENEMAS AXIOMS CC BY-NC-ND 3.0. Revised: 18
Nov 2012
57. Section 13 Incidence Geometry We will use the expression a
geometry to refer to the consequences of a particular set of
axioms. For example by Hilbert Geometry we mean the geometry that
is the consequence of Hilberts axioms; by Euclidean Geometry we
mean the consequences of Euclids Axioms, etc. Here we will describe
a particular type of nite geometry, that is, a geometry with a nite
number of points. Incidence Geometry is a term we will use for the
geometry that we can derive from the following three axioms. Axiom
13.1 (Incidence Axiom 1) For every pair of distinct points P and Q
there exists exactly one line such that both P and Q lie on . Axiom
13.2 (Incidence Axiom 2) For every line there exists at least two
distinct points P and Q such both P and Q lie on . Axiom 13.3
(Incidence Axiom 3) There exist three points that do not all lie on
the same line, i.e., there exists three non-collinear points.
Denition 13.4 Three points A, B, C are collinear if there exists at
least one line such that all three points line on . They are said
to be non- collinear if no such line exists. Example 13.1 3-Point
Plane Geometry. In this example of a nite geometry that obeys all
the axioms of incidence geometry, we dene: A point is an element of
the set {A, B, C} A line is a pair of points such as = {A, B} A
point P lies on a line if P . 53
58. 54 SECTION 13. INCIDENCE GEOMETRY Figure 13.1: Graph
diagram illustrating three-point geometry. Parallel lines do not
exist in this geometry. There are three possible lines in this
geometry: {A, B}, {B, C}, {A, C} We will describe nite geometries
with graph-diagrams consisting of nodes and line segments (gure
13.1). The nodes represent the points, and the line segments
connecting the nodes represent the sets that represent lines. This
is a representation of a nite geometry, not a picture of the points
of lines in the usual sense. In other words, the 3-point geometry
does not look like a triangle; we just represent it by a graph that
looks like a triangle. Example 13.2 Four-point Geometry. Here we
dene (see gure 13.2): A point is an element of the set {A, B, C, D}
A line is a pair of points such as = {A, B} A point P lies on a
line if P . There are six lines in this example: {A, B}, {A, C},
{A, D}, {B, C}, {B, D}, and {C, D}. Example 13.3 Fanos Geometry.
Here we have seven points given by the set {A, B, C, D, E, F, G}
and we dene lines as any of the following seven specic subsets: {A,
B, C}, {C, D, E}, {E, F, A}, {A, G, D}, {C, G, F}, {E, G, B}, {B,
D, F} as illustrated in gure 13.3 CC BY-NC-ND 3.0. Revised: 18 Nov
2012
59. SECTION 13. INCIDENCE GEOMETRY 55 Figure 13.2: Graph
diagram illustrating four-point geometry. Every line has precisely
one other line that is parallel to it, and and there is precisely
one parallel line through each point that is not on a given line,
and hence four point geometry obeys the Euclidean parallel
postulate. Figure 13.3: Graph diagram illustrating Fanos geometry.
Each line seg- ment and the circle represents a line in this
geometry. Revised: 18 Nov 2012 CC BY-NC-ND 3.0.
60. 56 SECTION 13. INCIDENCE GEOMETRY Example 13.4 The
Cartesian Plane. This is the traditional example we use in geometry
and it obeys all the rules of incidence geometry. Dene a point as
an ordered pair of real numbers {x, y}. Then a line is the
collection of all points ax + by + c = 0 for some choice of real
numbers a, b, c. The usual notation for this set is R2 . Example
13.5 Surface of a Sphere. Consider a unit sphere centered at the
origin in normal 3-space. The surface of this sphere is given by
the set of all points {x, y, z} such that x2 + y2 + z2 = 1 Dene a
point as any point on the surface of the sphere, and dene a line as
any great circle on the plane (a great circle is the intersection
of the unit sphere with any plane that goes through the center of
sphere; or equivalently, it is any circle on the sphere whose
radius is equal to the radius of the sphere, which is 1). This
geometry does not obey incidence geometry because any two antipodal
points (points at opposite poles of the sphere) are on an innite
number of common lines. This violates Incidence Axiom 1.
Furthermore, there are no parallel lines in this geometry because
any two great circles meet. Example 13.6 The Klein Disk. Consider
the interior of the unit disk centered at the origin of R2 . This
is the set of all points such that x2 + y2 < 1 Dene a point as
any ordered pair of numbers (x, y) such that x2 +y2 < 1, and
dene a line as any the part of any Euclidean line that lies inside
this circle. See gure 13.4. The Klein Disk obeys incidence geometry
but does not obey Euclids fth postulate. Denition 13.5 (Parallel
Lines) Two lines and m are said to be parallel if there is no point
P such that P lies on both and m. We denote this by m. Euclids fth
axiom is equivalent to the following statement: Axiom 13.6
(Euclidian Parallel Postulate) For every line and for every point P
there is exactly one line m such that P lies on m and m . Four
point geometry and geometry of the Cartesian plane each satisfy the
Euclidean Parallel Postulate. There are two other possible parallel
postulates that are incompatible with the Euclidean Parallel
Postulate but which lead to consistent geometries. CC BY-NC-ND 3.0.
Revised: 18 Nov 2012
61. SECTION 13. INCIDENCE GEOMETRY 57 Figure 13.4: A Klein
Disk. Lines l and n pass through point P, and are both parallel to
line m. The Klein Disk is a model of Hyperbolic Geometry. Axiom
13.7 (Elliptic Parallel Postulate) For every line and for every
point P there is no line m such that P lies on m and m .
Three-point geometry and geometry on the sphere satisfy the
Elliptic par- allel postulate. Axiom 13.8 (Hyperbolic Parallel
Postulate) For every line and for every point P there are at least
two lines m such that P lies on m and m . The Klein Disk and Five
Point geometry (gure 13.5) satisfy the hyperbolic parallel
postulate. Theorem 13.9 If and m are distinct, nonparallel lines,
then there exists a unique point P such that P lines on both and m.
Proof. By hypothesis, = m and m. Then by the negation of the
denition of parallel lines, there is a point P that lines on both
and m. To proove that P is unique, we assume that there is a second
point Q = P that also lies on both lines as our RAA hypothesis. By
incidence axiom 1, there is exactly one line n that contains both P
and Q. Since P is on , then since n is unique, = n. But since Q is
on m, then since n is unique, m = n. Hence = m. This contradicts
the hypothesis that = m. Hence our RAA Revised: 18 Nov 2012 CC
BY-NC-ND 3.0.
62. 58 SECTION 13. INCIDENCE GEOMETRY Figure 13.5: Illustration
of ve point geometry: the points are represented by the symbols A,
B, C, D.E and the lines are represented by any subset of precisely
two points. Since lines {B, D} and {B, C} are both parallel to line
{A, E} and both pass through point B, this model satises the
hyperbolic parallel postulate hypothesis must be wrong, i.e., P =
Q. Hence the point P is unique. Here are some other useful results
that hold in incidence geometry. Theorem 13.10 Let be a line. Then
there exists at least one point P that does not lie on . Theorem
13.11 Let P be a point. Then there exist at least two distinct
lines that contain P. Proof. By Incidence Axiom 3 there exist at
least three points R, S, T that are non-collinear. Either P {R, S,
T} or P {R, S, T}. If P {R, S, T} then without loss of generality
we can relabel the points so that P = R. Dene = PS and m = PT.
Since P, S, T are non-collinear, then and m are the desired lines.
If P {R, S, T} then either P RS or P RS. If P RS then let = RS and
m = PT. By construction, T RS, else R, S, T would be collinear.
Since T m and T , = m. Hence and m are the two distinct lines that
contain P. If P RS then let = PR and m = PS. By construction R ,
but CC BY-NC-ND 3.0. Revised: 18 Nov 2012
63. SECTION 13. INCIDENCE GEOMETRY 59 R m, else P, R, S would
be collinear, and we have assumed otherwise. Hence and m are
distinct lines that contain P. Theorem 13.12 Let be a line. Then
there exist two distinct lines m and n that intersect . Proof. Let
be a line.By incidence axiom 2 there are two points P, Q that lie
on . By theorem 13.10 there exists a third point R that does lie on
. By incidence axiom 1, there exist lines m = PR and n = QR. Since
P and P m, intersects m (denition of intersection). Since Q and Q
n, intersects n (denition of intersection). Since R then any line
that contains R is dierent from . The two lines m and n contain R,
hence m = and n = . Suppose m = n. By denition of m, P m. By
dention of n, Q n. Hence if m = n, Q m, i.e., both P and Q line on
m. Hence m = by the uniqueness part of incidence axiom 1. This
contradicts the previous paragraph, m = . Hence the assumption m =
n must be wrong. Hence m = n, which means there are two distinc
lines that intersect . Theorem 13.13 Let P be a point. Then there
exists at least one line that does not contain P. Theorem 13.14
There exist three distinct lines such that no point lies on all
three of them. Theorem 13.15 Let P be a point. Then there exist
points Q and R such that P, Q, R are non-collinear. Proof. Let P
and Q be points such that Q = P. By incidence axiom 1, there is a
unique line that contains P and Q. By theorem 13.10 there exists at
least one point R that does lie on . The points P, Q, R are
non-collinear. Theorem 13.16 Let P = Q be points. Then there exists
a point Q such that P, Q, R are non-collinear. Revised: 18 Nov 2012
CC BY-NC-ND 3.0.
64. 60 SECTION 13. INCIDENCE GEOMETRY CC BY-NC-ND 3.0. Revised:
18 Nov 2012
65. Section 14 Betweenness In this section we will begin our
formulation of plane geometry based on Venemas axiomatic system.
Our undened terms are: point, line, distance, half-plane, and angle
measure. Axiom 14.1 (Existence Postulate) The collection of
all
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