# A Foundation for Geometry

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 MODULE 4 A Foundation for Geometry There is no royal road to geometry Euclid 1. Points and Lines We are ready (finally!) to talk about geometry. Our first task is to say something about points and lines. We will assume that we have a (nonempty) space X and that the points in our geometry are just elements of X. There are no restrictions (for now) on the set X. It could be a finite set, the real number line, the x y plane, or even three dimensional space. We assume further that we have a set L of subsets of X. The elements of L are called the lines of our geometry. As with metric spaces we need some axioms or rules about the possible points and lines of our geometry. The first three rules are called the incidence axioms. They are very basic. Incidence Axioms Axiom 1. There are at least two different lines in L. Axiom 2. Each line contains at least two different points. Axiom 3. For any two distinct points A and B, there is exactly one line l in L which contains them. The third axiom is often stated as two points determine a line and dates back to Euclid. The unique line determined by A and B is written AB. Points which lie on the same line are called collinear. Our first theorem in geometry is an easy consequence of Axiom 3 Theorem 4.1. Distinct lines m and n intersect in at most one point 71

2 72 4. A FOUNDATION FOR GEOMETRY 4.1 Exercises 1. Prove Theorem Prove that there must be at least three points in X. 2. Distance and Betweenness We will assume that our space has a distance function d defined on it which obeys the four distance axioms of a metric space. It is convenient to use a simplified notation for the distance between two points. Assuming that we are using the distance function d, then we shall denote the distance d(a,b) between two points A and B by AB def = d(a,b). We use the topological axioms of distance as a springboard into geometry. In terms of this simplified notation, the four distance axioms can be stated. Distance Axioms Axiom 1. AB 0 [non negativity] Axiom 2. AB = 0 if and only if A = B [zero property] Axiom 3. AB = BA [symmetry] Axiom 4. AC AB +BC [triangle inequality] Remark. Most authors do not include Axiom 4, the triangle inequality, since it is possible to prove the triangle inequality from the other axioms of geometry. Since we are not critically interested in developing a minimal set of axioms here, it suits our purpose to assume a priori that our distance function d satisfies the triangle inequality. Given three distinct points A, B, or C on a line, we often need to know in which order they appear. In general we expect that one of them will lie between the other two. But the question is: which one is in the middle? The Greek geometers did not define the concept of betweenness, relying on diagrams instead.

3 3. EXAMPLES OF GEOMETRIES 73 We will use the notation A B C to designate that A, B, and C are collinear points and that B is between A and C. We can use distance to define betweenness, as follows: Definition 4.1. A B C if and only if A, B, C are three distinct collinear points and AB +BC = AC. The following theorem formalizes the intuitively obvious statement that if three points line up in order A,B,C, then going backwards they are in order C,B,A. Theorem 4.2. A B C implies C B A. The definition of betweenness does not assert that a betweenness relation must hold between three collinear points. It merely defines the condition that must be met if a betweenness relation actually occurs. The existence of a betweenness relation requires another axiom. Betweenness Axiom. Given three distinct collinear points. At least one betweenness relation A B C, A C B, or B A C holds. The next theorem says that only one of the points A, B, or C can lie in the middle. For this reason, we call this the unique middle theorem. Theorem 4.3. [unique middle] If A B C is true, then A C B and B A C are both false. 4.2 Exercises 1. Prove Theorem Prove Theorem Examples of Geometries Example 4.1. [The Euclidean plane] Let the space X be the x y plane {(x,y) : x and y are real numbers}. The lines in L are sets of the form {(x,y) : ax+by = c},

4 74 4. A FOUNDATION FOR GEOMETRY where a, b, and c are real numbers, not all zero. The usual definition of distance between two points P = (x 1,y 1 ) and Q = (x 2,y 2 ) in the plane is d(p,q) = (x 1 x 2 ) 2 +(y 1 y 2 ) 2. Example 4.2 (The x y plane with the taxicab metric). X and L are defined as in Example 4.1. The distance between two points P = (x 1,y 1 ) and Q = (x 2,y 2 ) in the plane is d(p,q) = x 1 x 2 + y 1 y 2. Example 4.3 (Lines on a sphere). Let X be a sphere and suppose that we restrict travel to the surface of the sphere. Then the distance between two points P and Q on the sphere is measured along the great circle of the sphere which passes through the two points. A great circle is a circle on the sphere whose center is the center of the sphere. Note that because of a technicality, the third incidence axiom does not hold on a sphere. Specifically, if A is the North Pole and B is the South Pole, then any longitudinal line passes through these two points. It is possible to modify the incidence axioms to allow for this exception. Example 4.4 (Discrete space). Let X be any set with at least three elements. The set L consists of all possible two element sets {a,b} where a and b lie in X. The following function is a distance function: d(a,b) = { 0 if a = b 1 if a b. Note that it is impossible to find three distinct points on any line in this model, so no betweenness relations hold. Example 4.5 (The Fano Plane). The Fano plane F is a collection of 7 points X = {A,B,C,D,E,F,G} and 7 lines, determined by the drawing in Figure 1. The seven lines are (1) A,B,D (2) A,C,E (3) B,C,F (4) A,F,G (5) B,E,G (6) C,D,G (7) D,E,F

5 3. EXAMPLES OF GEOMETRIES 75 A D E G B F Figure 1. The Fano plane C The solid and dotted lines in the figure are only to aid visualization of the lines. There are only seven points and seven lines in the Fano plane. The Fano plane is also called the projective plane of order 2. Distances are assigned so that the distance function satisfies the zero property and symmetry, that is, the distance from a point to itself is 0 and the distance from point x to point y is the same as the distance from point y to point x. Six distances are defined to be 2: AB = AC = AF = BE = BC = CD = 2 The remaining distances d(x,y), when x y, have the value Exercises 1. On a sphere of radius 13, what is the distance between A = (0,0,13) and (3,4,12)? 2. We have defined lines in the x y plane with the standard distance function to be the same as lines in the x y plane using the taxicab metric. Show that if A, B, and C are three distinct points on a line in the x y plane, then the betweenness relation A B C holds using the standard distance function if and only if A B C holds using the taxicab metric. 3. What are the betweenness relations for the discrete space?

6 76 4. A FOUNDATION FOR GEOMETRY 4. Ignoring symmetry, there are six betweenness relations in the Fano Plane. Find them. 5. Show that the Fano plane F has the following interesting property: Property (*): Any line passing through any two points in F passes through a third point in F. You will never find a finite collection of non-collinear points in the Euclidean x y plane that satisfy Property (*). 6. The Fano plane has a remarkable coloring property. To warm up on an easier example, consider the following three pairs of letters: (1) {A,B}, (2) {A,C}, and (3) {B,C}. Now suppose you color each letter, using exactly two colors, red and blue. Each letter must be colored the same in every pair it appears. For example, if you color A in the first pair red, then you must color A in the second pair red also. Explain why, no matter how the three letters are colored, at least one of the three pairs will have the same color. Suppose we switch from pairs to triples. Can we construct an example similar to the three pairs dislayed above? The key is to use the Fano plane. Show that no matter how you color the seven points in the Fano plane F using two colors (red and blue), it will always be the case that at least one of the seven lines will contain three points all of the same color. 4. Segments and Rays We can use the notion of betweenness to define segments and rays in our geometry. Suppose X is a set of points, d is a distance function on X, and L is the set of lines. Definition 4.2 (line segment AB). If A and B are distinct points in X, then AB def = {A,B} {P : A P B}. The points A and B are called the endpoints of the segment AB. A segment consists of its endpoints and the points in between them. The points strictly between A and B are called the interior points of AB. Definition 4.3 (ray AB). If A and B are distinct points in X, then AB def = AB {P : A B P}.

7 4. SEGMENTS AND RAYS 77 The starting point A is called the endpoint of ray AB. The points on AB other than A are called the interior points of AB. The following three theorems, which are intuitively obvious if you draw diagrams, can be proven readily using our definitions of line segment and ray. Theorem 4.4. AB = BA Theorem 4.5. AB BA = AB Theorem 4.6. AB BA = AB We would like to have an axiom which says that our rays look like rays on the real numbers line. This is accomplished by the following axiom. The Ruler Axiom. Given a ray AB and a real number t 0. Then there is a point X on AB such that AX = t. The Ruler Axiom can be used to prove the following theorem about midpoints: Theorem 4.7. [midpoint] Given segment AB there is a unique point M on AB, called the midpoint of AB, such that AM = MB. 4.4 Exercises 1. Write an equation of the form y = mx + b to describe the segment AB in R 2 with A = (0,0) and B = (1,2). 2. Let l be the line y = 2x+1 in the x y plane. (a) Find two point on l whose Euclidean distance from the point (0,1) is precisely 5. (b) Find two point on l whose taxicab distance from the point (0,1) is precisely Is it possible to find two points X 1 and X 2 on AB whose distance from A is t? 4. Is it possible to find two points X 1 and X 2 on AB whose distance from A is t? 5. Prove Theorem 4.4. Hint: Use Theorem Prove Theorem 4.5. Hint: Use Theorem Prove Theorem 4.6. Hint: Use the Betweenness Axiom.

8 78 4. A FOUNDATION FOR GEOMETRY 8. Prove Theorem Dimension If we have a betweenness relation B A C then the rays AB and AC are called opposite rays. Notice that the ray AB together with the opposite ray AC comprise the entire line AB. Since the two rays meet only at the endpoint A, we see that a point A on a line l separates the line into three disjoint pieces: 1. the point A itself 2. the interior of the ray AB lying on l 3. the interior of the opposite ray AC lying on l. In a similar way a line l separates the plane into three disjoint pieces: 1. the line l itself 2. a halfplane on one side of l 3. the opposite halfplane on the other side of l. This separation fact is one of the basic axioms used in studying the foundations of geometry. Rather than try to state a precise definition of the separation axiom, we will just contend ourselves with this informal description. It is separation by lines that makes our geometry two dimensional. Consider the three dimensional world we live in. To locate a point in our world we must specify three values: width, depth, and height. All the other axioms of geometry hold in three dimensional space: two points determine a line; the Betweenness Axiom; the Ruler Axiom. Segments and rays can be defined in three space using the notion of betweenness. A line, however, does not separate three dimensional space into three disjoint pieces. Separtion in three space requires a plane, which acts like a wall, dividing the space into the wall itself and the two regions on either side of the wall. The idea of dimension works as follows. A point has dimension 0; it has no width, depth, or height. A line is one dimensional; it can be separated into three pieces by a 0 dimensional point. A plane has two dimensions; it can be separated by a one dimensional line. Three dimensional space is separated by a two dimensional plane.

9 5. DIMENSION Exercises 1. Consider the x y plane and take l to be the line y = 2x+1. Describe the two halfplanes separated by l. 2. Consider the sphere of the earth. What halfplanes are separated by the equator? 3. Given two random lines in three dimensional space, do you think it is very likely that they intersect? 4. Think of the fourth dimension as time. Explain how a three dimension subspace of space-time at time t = 0 separates space-time into three distinct parts.

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

### Foundations of Geometry 1: Points, Lines, Segments, Angles

Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.

### 4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem.

14 Perpendicularity and Angle Congruence Definition (acute angle, right angle, obtuse angle, supplementary angles, complementary angles) An acute angle is an angle whose measure is less than 90. A right

### Mathematics 3301-001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3

Mathematics 3301-001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3 The problems in bold are the problems for Test #3. As before, you are allowed to use statements above and all postulates in the proofs

### 12. Parallels. Then there exists a line through P parallel to l.

12. Parallels Given one rail of a railroad track, is there always a second rail whose (perpendicular) distance from the first rail is exactly the width across the tires of a train, so that the two rails

### GEOMETRY. Constructions OBJECTIVE #: G.CO.12

GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic

### DESARGUES THEOREM DONALD ROBERTSON

DESARGUES THEOREM DONALD ROBERTSON Two triangles ABC and A B C are said to be in perspective axially when no two vertices are equal and when the three intersection points AC A C, AB A B and BC B C are

### Introduction. Modern Applications Computer Graphics Bio-medical Applications Modeling Cryptology and Coding Theory

Higher Geometry Introduction Brief Historical Sketch Greeks (Thales, Euclid, Archimedes) Parallel Postulate Non-Euclidean Geometries Projective Geometry Revolution Axiomatics revisited Modern Geometries

### half-line the set of all points on a line on a given side of a given point of the line

Geometry Week 3 Sec 2.1 to 2.4 Definition: section 2.1 half-line the set of all points on a line on a given side of a given point of the line notation: is the half-line that contains all points on the

### alternate interior angles

alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

### MA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry. Figure 1: Lines in the Poincaré Disk Model

MA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry Put your name here: Score: Instructions: For this lab you will be using the applet, NonEuclid, created by Castellanos, Austin, Darnell,

### 3.1 Triangles, Congruence Relations, SAS Hypothesis

Chapter 3 Foundations of Geometry 2 3.1 Triangles, Congruence Relations, SAS Hypothesis Definition 3.1 A triangle is the union of three segments ( called its side), whose end points (called its vertices)

### Lesson 18: Looking More Carefully at Parallel Lines

Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using

### SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses

CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning

### Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment

Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points

### MODERN APPLICATIONS OF PYTHAGORAS S THEOREM

UNIT SIX MODERN APPLICATIONS OF PYTHAGORAS S THEOREM Coordinate Systems 124 Distance Formula 127 Midpoint Formula 131 SUMMARY 134 Exercises 135 UNIT SIX: 124 COORDINATE GEOMETRY Geometry, as presented

### Arrangements And Duality

Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,

### DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent

### Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

### Geometry: Euclidean. Through a given external point there is at most one line parallel to a

Geometry: Euclidean MATH 3120, Spring 2016 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list,

### INTRODUCTION TO EUCLID S GEOMETRY

78 MATHEMATICS INTRODUCTION TO EUCLID S GEOMETRY CHAPTER 5 5.1 Introduction The word geometry comes form the Greek words geo, meaning the earth, and metrein, meaning to measure. Geometry appears to have

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, June 19, :15 a.m. to 12:15 p.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 19, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### Chapter 4.1 Parallel Lines and Planes

Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about

### GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd:

GEOMETRY Chapter 1: Foundations for Geometry Name: Teacher: Pd: Table of Contents Lesson 1.1: SWBAT: Identify, name, and draw points, lines, segments, rays, and planes. Pgs: 1-4 Lesson 1.2: SWBAT: Use

### 1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?

1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width

### Geometry in a Nutshell

Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

### Solutions to in-class problems

Solutions to in-class problems College Geometry Spring 2016 Theorem 3.1.7. If l and m are two distinct, nonparallel lines, then there exists exactly one point P such that P lies on both l and m. Proof.

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

### Geometry Regents Review

Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

### Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)

Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer

### Solutions to Practice Problems

Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### Synthetic Projective Treatment of Cevian Nests and Graves Triangles

Synthetic Projective Treatment of Cevian Nests and Graves Triangles Igor Minevich 1 Introduction Several proofs of the cevian nest theorem (given below) are known, including one using ratios along sides

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### 55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.

Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit

### Mathematics Geometry Unit 1 (SAMPLE)

Review the Geometry sample year-long scope and sequence associated with this unit plan. Mathematics Possible time frame: Unit 1: Introduction to Geometric Concepts, Construction, and Proof 14 days This

### Unit 3: Triangle Bisectors and Quadrilaterals

Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties

### 5-1 Perpendicular and Angle Bisectors

5-1 Perpendicular and Angle Bisectors Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Construct each of the following. 1. A perpendicular bisector. 2. An angle bisector. 3. Find the midpoint and

### @12 @1. G5 definition s. G1 Little devils. G3 false proofs. G2 sketches. G1 Little devils. G3 definition s. G5 examples and counters

Class #31 @12 @1 G1 Little devils G2 False proofs G3 definition s G4 sketches G5 examples and counters G1 Little devils G2 sketches G3 false proofs G4 examples and counters G5 definition s Jacob Amanda

### Terminology: When one line intersects each of two given lines, we call that line a transversal.

Feb 23 Notes: Definition: Two lines l and m are parallel if they lie in the same plane and do not intersect. Terminology: When one line intersects each of two given lines, we call that line a transversal.

### Coordinate Plane Project

Coordinate Plane Project C. Sormani, MTTI, Lehman College, CUNY MAT631, Fall 2009, Project XI BACKGROUND: Euclidean Axioms, Half Planes, Unique Perpendicular Lines, Congruent and Similar Triangle Theorems,

### Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

### 2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

### Lesson 19: Equations for Tangent Lines to Circles

Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

### Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### Congruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key

Instruction Goal: To provide opportunities for students to develop concepts and skills related to identifying and constructing angle bisectors, perpendicular bisectors, medians, altitudes, incenters, circumcenters,

### CHAPTER 8, GEOMETRY. 4. A circular cylinder has a circumference of 33 in. Use 22 as the approximate value of π and find the radius of this cylinder.

TEST A CHAPTER 8, GEOMETRY 1. A rectangular plot of ground is to be enclosed with 180 yd of fencing. If the plot is twice as long as it is wide, what are its dimensions? 2. A 4 cm by 6 cm rectangle has

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### Classical theorems on hyperbolic triangles from a projective point of view

tmcs-szilasi 2012/3/1 0:14 page 175 #1 10/1 (2012), 175 181 Classical theorems on hyperbolic triangles from a projective point of view Zoltán Szilasi Abstract. Using the Cayley-Klein model of hyperbolic

### Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1

Student Name: Teacher: Date: District: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the

### 0810ge. Geometry Regents Exam 0810

0810ge 1 In the diagram below, ABC XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements identify

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 20, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.

Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### Chapter 23. The Reflection of Light: Mirrors

Chapter 23 The Reflection of Light: Mirrors Wave Fronts and Rays Defining wave fronts and rays. Consider a sound wave since it is easier to visualize. Shown is a hemispherical view of a sound wave emitted

### Geometry Module 4 Unit 2 Practice Exam

Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning

### Surface Area of Rectangular & Right Prisms Surface Area of Pyramids. Geometry

Surface Area of Rectangular & Right Prisms Surface Area of Pyramids Geometry Finding the surface area of a prism A prism is a rectangular solid with two congruent faces, called bases, that lie in parallel

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### Unknown Angle Problems with Inscribed Angles in Circles

: Unknown Angle Problems with Inscribed Angles in Circles Student Outcomes Use the inscribed angle theorem to find the measures of unknown angles. Prove relationships between inscribed angles and central

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### CHAPTER 6 LINES AND ANGLES. 6.1 Introduction

CHAPTER 6 LINES AND ANGLES 6.1 Introduction In Chapter 5, you have studied that a minimum of two points are required to draw a line. You have also studied some axioms and, with the help of these axioms,

### Projective Geometry - Part 2

Projective Geometry - Part 2 Alexander Remorov alexanderrem@gmail.com Review Four collinear points A, B, C, D form a harmonic bundle (A, C; B, D) when CA : DA CB DB = 1. A pencil P (A, B, C, D) is the

### 4. How many integers between 2004 and 4002 are perfect squares?

5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

### MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem

MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and

### PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b.

PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.

### The Geometer s Sketchpad: Non-Euclidean Geometry & The Poincaré Disk

The Geometer s Sketchpad: Non-Euclidean Geometry & The Poincaré Disk Nicholas Jackiw njackiw@kcptech.com KCP Technologies, Inc. ICTMT11 2013 Bari Overview. The study of hyperbolic geometry and non-euclidean

### 15. Appendix 1: List of Definitions

page 321 15. Appendix 1: List of Definitions Definition 1: Interpretation of an axiom system (page 12) Suppose that an axiom system consists of the following four things an undefined object of one type,

### This is a tentative schedule, date may change. Please be sure to write down homework assignments daily.

Mon Tue Wed Thu Fri Aug 26 Aug 27 Aug 28 Aug 29 Aug 30 Introductions, Expectations, Course Outline and Carnegie Review summer packet Topic: (1-1) Points, Lines, & Planes Topic: (1-2) Segment Measure Quiz

### 1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

### CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY

CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY Specific Expectations Addressed in the Chapter Explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

### Lesson 2: Circles, Chords, Diameters, and Their Relationships

Circles, Chords, Diameters, and Their Relationships Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct

### What is inversive geometry?

What is inversive geometry? Andrew Krieger July 18, 2013 Throughout, Greek letters (,,...) denote geometric objects like circles or lines; small Roman letters (a, b,... ) denote distances, and large Roman

### Math 3372-College Geometry

Math 3372-College Geometry Yi Wang, Ph.D., Assistant Professor Department of Mathematics Fairmont State University Fairmont, West Virginia Fall, 2004 Fairmont, West Virginia Copyright 2004, Yi Wang Contents

### Intermediate Math Circles October 10, 2012 Geometry I: Angles

Intermediate Math Circles October 10, 2012 Geometry I: Angles Over the next four weeks, we will look at several geometry topics. Some of the topics may be familiar to you while others, for most of you,

### Trigonometric Functions and Triangles

Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### Elements of Plane Geometry by LK

Elements of Plane Geometry by LK These are notes indicating just some bare essentials of plane geometry and some problems to think about. We give a modified version of the axioms for Euclidean Geometry

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### GEOMETRY FINAL EXAM REVIEW

GEOMETRY FINL EXM REVIEW I. MTHING reflexive. a(b + c) = ab + ac transitive. If a = b & b = c, then a = c. symmetric. If lies between and, then + =. substitution. If a = b, then b = a. distributive E.

### 37 Basic Geometric Shapes and Figures

37 Basic Geometric Shapes and Figures In this section we discuss basic geometric shapes and figures such as points, lines, line segments, planes, angles, triangles, and quadrilaterals. The three pillars

### Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18

Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

### PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin

PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles

### Tangent circles in the hyperbolic disk

Rose- Hulman Undergraduate Mathematics Journal Tangent circles in the hyperbolic disk Megan Ternes a Volume 14, No. 1, Spring 2013 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics

### Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. 1. measures less than By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than

### CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

### Inversion in a Circle

Inversion in a Circle Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles March 23, 2011 Abstract This article will describe the geometric tool of inversion in a circle, and will demonstrate