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Geometry I MTH-2007-2 MTH-2007-2 GEOMETRY I This course was produced in collaboration with the Service de l'éducation des adultes de la Commission scolaire catholique de Sherbrooke and the State Secretary of Canada. Author: Marie-Reine Rouillard Content revision: Jean-Paul Groleau Diane Vigneux Linguistic revision: Kay Flanagan and Leslie Macdonald Consultant in andragogy: Serge Vallières Coordinator for the DGFD: Jean-Paul Groleau Coordinator for the DGEA: Ronald Côté Photocomposition and layout: Multitexte Plus Translation: Consultation en éducation Zegray First Edition: 1991 Reprint: 2001 © Société de formation à distance des commissions scolaires du Québec All rights for translation and adaptation, in whole or in part, reserved for all countries. Any reproduction, by mechanical or electronic means, including micro-reproduction, is forbidden without the written permission of a duly authorized representative of the Société de formation à distance des commissions scolaires du Québec (SOFAD). Legal Deposit – 2001 Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada ISBN 978-2-89493-198-1 MTH-2007-2 Geometry I TABLE OF CONTENTS of: This is a preview n; and Introduction to the Program Flowchart ................................................... 0.4 - the introductio it. The Program Flowchart ............................................................................ 0.5 - the first un How to Use this Guide .............................................................................. 0.6 General Introduction ................................................................................. 0.9 Intermediate and Terminal Objectives of the Module ............................ 0.10 Diagnostic Test on the Prerequisites ....................................................... 0.15 Answer Key for the Diagnostic Test on the Prerequisites ...................... 0.19 Analysis of the Diagnostic Test Results ................................................... 0.23 Information for Distance Education Students ......................................... 0.25 UNITS 1. 2. 3. 4. 5. 6. 7. 8. Basic Geometric Concepts......................................................................... 1.1 Drawing an Angle ..................................................................................... 2.1 Types of Lines ............................................................................................ 3.1 Categories of Angles .................................................................................. 4.1 Polygons ..................................................................................................... 5.1 Measuring Polygons .................................................................................. 6.1 Pythagoras' Theorem ................................................................................ 7.1 Special Right Triangles and Pythagoras' Theorem ................................. 8.1 Final Review .............................................................................................. 9.1 Terminal Objectives .................................................................................. 9.5 Self-Evaluation Test.................................................................................. 9.7 Answer Key for the Self-Evaluation Test ................................................ 9.13 Analysis of the Self-Evaluation Test Results .......................................... 9.17 Final Evaluation........................................................................................ 9.18 Answer Key for the Exercises ................................................................... 9.19 Glossary ..................................................................................................... 9.71 List of Symbols .......................................................................................... 9.76 Bibliography .............................................................................................. 9.77 Review Activities ..................................................................................... 10.1 © SOFAD 0.3 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I INTRODUCTION TO THE PROGRAM FLOWCHART WELCOME TO THE WORLD OF MATHEMATICS This mathematics program has been developed for adult students enrolled either with Adult Education Services of school boards or in distance education. The learning activities have been designed for individualized learning. If you encounter difficulties, do not hesitate to consult your teacher or to telephone the resource person assigned to you. The following flowchart shows where this module fits into the overall program. It allows you to see how far you have come and how much further you still have to go to achieve your vocational objective. There are three possible paths you can take, depending on your goal. The first path, which consists of Modules MTH-3003-2 (MTH-314) and MTH-4104-2 (MTH-416), leads to a Secondary School Vocational Diploma (SSVD). The second path, consisting of Modules MTH-4109-1 (MTH-426), MTH-4111-2 (MTH-436) and MTH-5104-1 (MTH-514), leads to a Secondary School Diploma (SSD), which gives you access to certain CEGEP programs that do not call for a knowledge of advanced mathematics. Lastly, the path consisting of Modules MTH-5109-1 (MTH-526) and MTH-5111-2 (MTH-536) will lead to CEGEP programs that require a thorough knowledge of mathematics in addition to other abilities. Good luck! If this is your first contact with the mathematics program, consult the flowchart on the next page and then read the section “How to Use this Guide.” Otherwise, go directly to the section entitled “General Introduction.” Enjoy your work! 0.4 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I THE PROGRAM FLOWCHART CEGEP MTH-5112-1 MTH-5111-2 MTH-536 MTH-5104-1 MTH-5103-1 Introduction to Vectors MTH-5109-1 Geometry IV MTH-5108-2 Trigonometric Functions and Equations MTH-5107-2 Exponential and Logarithmic Functions and Equations Optimization II MTH-5106-1 Real Functions and Equations Probability II MTH-5105-1 Conics MTH-5102-1 Statistics III MTH-5101-1 Optimization I MTH-4111-2 Trades DVS MTH-436 MTH-4110-1 MTH-4109-1 MTH-426 MTH-314 MTH-216 MTH-116 © SOFAD The Four Operations on Algebraic Fractions Sets, Relations and Functions MTH-4108-1 Quadratic Functions Straight Lines II MTH-4106-1 Factoring and Algebraic Functions MTH-4105-1 Exponents and Radicals MTH-4103-1 MTH-4102-1 MTH-4101-2 Complement and Synthesis I MTH-4107-1 MTH-4104-2 MTH-416 Complement and Synthesis II MTH-5110-1 MTH-526 MTH-514 Logic Statistics II Trigonometry I Geometry III Equations and Inequalities II MTH-3003-2 Straight Lines I MTH-3002-2 Geometry II MTH-3001-2 The Four Operations on Polynomials MTH-2008-2 Statistics and Probabilities I MTH-2007-2 Geometry I MTH-2006-2 Equations and Inequalities I You ar e h er e MTH-1007-2 Decimals and Percent MTH-1006-2 The Four Operations on Fractions MTH-1005-2 The Four Operations on Integers 0.5 25 hours = 1 credit 50 hours = 2 credits ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I HOW TO USE THIS GUIDE Hi! My name is Monica and I have been asked to tell you about this math module. What’s your name? Whether you are registered at an adult education center or at Formation à distance, ... Now, the module you have in your hand is divided into three sections. The first section is... I’m Andy. ... you have probably taken a placement test which tells you exactly which module you should start with. ... the entry activity, which contains the test on the prerequisites. 0.6 You’ll see that with this method, math is a real breeze! My results on the test indicate that I should begin with this module. By carefully correcting this test using the corresponding answer key, and recording your results on the analysis sheet ... © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I ... you can tell if you’re well enough prepared to do all the activities in the module. And if I’m not, if I need a little review before moving on, what happens then? In that case, before you start the activities in the module, the results analysis chart refers you to a review activity near the end of the module. I see! In this way, I can be sure I have all the prerequisites for starting. START The starting line shows where the learning activities begin. Exactly! The second section contains the learning activities. It’s the main part of the module. ? The little white question mark indicates the questions for which answers are given in the text. The target precedes the objective to be met. The memo pad signals a brief reminder of concepts which you have already studied. ? Look closely at the box to the right. It explains the symbols used to identify the various activities. The boldface question mark indicates practice exercices which allow you to try out what you have just learned. The calculator symbol reminds you that you will need to use your calculator. ? The sheaf of wheat indicates a review designed to reinforce what you have just learned. A row of sheaves near the end of the module indicates the final review, which helps you to interrelate all the learning activities in the module. FINISH Lastly, the finish line indicates that it is time to go on to the self-evaluation test to verify how well you have understood the learning activities. © SOFAD 0.7 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I There are also many fun things in this module. For example, when you see the drawing of a sage, it introduces a “Did you know that...” It’s the same for the “math whiz” pages, which are designed especially for those who love math. For example. words in boldface italics appear in the glossary at the end of the module... A “Did you know that...”? Yes, for example, short tidbits on the history of mathematics and fun puzzles. They are interesting and relieve tension at the same time. Must I memorize what the sage says? No, it’s not part of the learning activity. It’s just there to give you a breather. They are so stimulating that even if you don’t have to do them, you’ll still want to. And the whole module has been arranged to make learning easier. ... statements in boxes are important points to remember, like definitions, formulas and rules. I’m telling you, the format makes everything much easier. The third section contains the final review, which interrelates the different parts of the module. Great! There is also a self-evaluation test and answer key. They tell you if you’re ready for the final evaluation. Thanks, Monica, you’ve been a big help. I’m glad! Now, I’ve got to run. See you! 0.8 Later ... This is great! I never thought that I would like mathematics as much as this! © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I GENERAL INTRODUCTION GEOMETRY: HERITAGE FROM ANCIENT TIMES According to a tradition which is over 3 000 years old, the origin of geometry (from the Greek "geo-" meaning earth and "metron" meaning measure) is attributed to an Egyptian pharaoh who hired "land measurers" to re-establish land boundaries along the shores of the Nile after the spring floods. Since the fences and boundary markers were destroyed yearly by the rising waters, the pharaoh had the lands measured each year so that he could readjust the taxes. It is thought that Nature probably provided the inspiration for a number of geometric concepts. Geometric figures occur everywhere in nature: leaves, flowers, spider webs, the honeycombs of a beehive, the wings of butterflies, snowflakes (always hexagonal in shape), and so on. Nature also provides examples of most of the geometric elements or figures studied in this module: the point, the line, the ray, the line segment, the right, acute, straight or obtuse angle, the equilateral or isosceles triangle, the isosceles right or scalene triangle, the parallelogram, the rhombus, the square and the trapezoid. To be in a position to draw geometric figures, you should have the appropriate instruments on hand. Your geometry set should include: a ruler graduated in centimetres and millimetres, a right triangle, a protractor and a compass. A description of these instruments is given in the first unit. The concept of line is covered in detail to enable you to distinguish intersecting lines (such as street intersections: X), perpendicular lines (such as antenna in the shape of a T), and parallel lines (such as railroad tracks: =). In addition, one unit deals with the different categories of pairs of angles and their characteristics. In it you will learn about complementary and supplementary angles, adjacent angles, vertically opposite angles, alternate exterior and alternate interior angles and finally, corresponding angles. The last activity involves studying Pythagoras' theorem and its applications to various right triangles. © SOFAD 0.9 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I INTERMEDIATE AND TERMINAL OBJECTIVES OF THE MODULE Module MTH-2007-2 (GSM-222)* contains seven units and requires fifty hours of study distributed as shown below. Each unit covers either an intermediate or a terminal objective. The terminal objectives appear in boldface. Objectives Number of Hours** % (evaluation) 10 20% 12 20% 13 30% GSM-222-07 8 20% GSM-222-08 5 10% GSM-222-01 and GSM-222-02 GSM-222-03 and GSM-222-04 GSM-222-05 and GSM-222-06 * GSM stands for “General Education, Secondary-Level, Mathematics.” ** Two hours are allotted for the final evaluation. 0.10 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 GSM-222-01 Geometry I Basic Geometric Concepts To distinguish between the following geometric figures: line, ray, line segment, angle, acute angle, obtuse angle, right angle, straight angle. You will have to distinguish between two figures that have already been drawn. You must also know how to measure a given angle between 0° and 180° to the nearest 2°, using a protractor. GSM 222-02 Drawing an Angle To draw an angle of n degrees to the nearest 2°, using a protractor. The measure of the angle to be drawn is a whole number between 0° and 180°. GSM 222-03 Types of Lines To distinguish between the following pairs of lines: parallel lines, perpendicular intersecting lines and non-perpendicular intersecting lines. GSM 222-04 Categories of Angles To determine the measure of one or more angles in a geometric figure, given the measure of one of its angles. The measure(s) will be determined by applying the properties of the following pairs of angles: complementary angles, supplementary angles, adjacent angles, vertically opposite angles, alternate interior angles, alternate exterior angles, corresponding angles. The use of a protractor is not permitted. © SOFAD 0.11 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 GSM 222-05 Geometry I Polygons Given a set of geometric figures illustrating polygons, to recognize those which represent triangles, equilateral triangles, isosceles triangles, right triangles, isosceles right triangles, scalene triangles, quadrilaterals, parallelograms, rhombuses, squares, rectangles and trapezoids. These figures will be identified by applying the properties of the angles, sides and diagonals in each of these polygons. The use of a protractor and a ruler is permitted. GSM 222-06 Measuring Polygons To determine the measures of angles and sides in a geometric figure consisting of various polygons: equilateral triangle, isosceles triangle, right triangle, isosceles right triangle, scalene triangle, parallelogram, rhombus, square, rectangle and trapezoid. The measures will be determined by applying the properties of the angles, sides and diagonals in these polygons. The measure of one or more angles or of one or more sides is indicated on the figure. The use of a protractor is not permitted. GSM 222-07 The Pythagoras' Theorem Given the measures of two sides of a right triangle, to apply Pythagoras' theorem in order to calculate the measure of the third side. The triangles used illustrate situations in everyday life. The steps in the solution must be shown. 0.12 © SOFAD TABLE OF CONTENTS ANSWER KEY MTH-2007-2 GSM 222-08 Geometry I Special Right Triangles and Pythagoras' Theorem Given the measure of one side of a right triangle where one of the angles measures 30° or 45°, to apply Pythagoras' theorem in order to calculate the measure of one of the other two sides. The triangles illustrate situations in everyday life. The steps in the solution must be shown. © SOFAD 0.13 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I DIAGNOSTIC TEST ON THE PREREQUISITES Instructions 1. Answer as many questions as you can. 2. Do not use a calculator. 3. Write your answers on the test paper. 4. Do not waste time. If you cannot answer a question, go on to the next one immediately. 5. When you have answered as many questions as you can, correct your answers, using the answer key which follows the diagnostic test. 6. To be considered correct, your answers must be identical to those in the key. For example, if you are asked to describe the steps involved in solving a problem, your answer must contain all the steps. 7. Transcribe your results onto the chart which follows the answer key. It gives an analysis of the diagnostic test results. 8. Do only the review activities which are suggested for each of your incorrect answers. 9. If all your answers are correct, you may begin working on this module. © SOFAD 0.15 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I 1. Perform the following additions. a) 1054.6 + 956.31 = .............................. because 1054.6 b) 3 + 1 = 7 2 ....................................................................................................... 2. Perform the following subtractions. a) 861 – 456.3 = ..................................... because 861 – b) 5 – 1 = 9 4 ....................................................................................................... 3. Perform the following multiplications. a) 5.6 × 3.15 = ........................................ because 3.15 × b) 3 × 5 = 8 11 ..................................................................................................... 4. Perform the following divisions. a) 53.55 ÷ 6.3 = ...................................... because 5355 b) 2 ÷ 1 = 3 7 ....................................................................................................... 0.16 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I 5. Solve each of the equations below. The complete solution is required. a) x + 4 = 9 b) 3x – 1 = 8 c) 4x – 2 = 7 9 3 d) 2(x + 4) = 17 6. Solve each of the following problems. The complete solution is required. a) John has 50 marbles. He gives 17 to his friend Josie. She thanks him and goes to find her other friend, Sophia, who gives her some more marbles. If Josie was given 23 marbles in total that afternoon, how many marbles did she get from Sophia? © SOFAD 0.17 TABLE OF CONTENTS MTH-2007-2 ANSWER KEY Geometry I b) Joey has a number of debts: he owes Mathilda $37.50, Gilbert $43.00 and Corinna $18.75. Unfortunately, he only has $67.42 in his bank account. Aware of his financial situation, his father offers to hire him at twice the salary he presently earns. If this represents an amount of $357.50 per week, what is Joey's present salary? c) Ms Leddy wants to sew new drapes for one of the windows of her cottage. Since the height of this window is only 0.5 of a metre, the drapes will not be very long! Ms Leddy also wishes to make four cushions with the fabric left over from the drapes. She estimates that she will need 6.5 metres of fabric in all. This quantity is triple the width of the window (for the drapes) plus 2 metres (for the cushions). What is the width of the window in Ms Leddy's cottage? 0.18 © SOFAD TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I ANSWER KEY FOR THE DIAGNOSTIC TEST ON THE PREREQUISITES 1. a) 1 054.6 + 956.31 = 2 010.91 because b) 3 + 1 = 6 + 7 = 13 7 2 14 14 14 2. a) 861 – 456.3 = 404.7 because b) 5 – 1 = 20 – 9 = 11 9 4 36 36 36 3. a) 5.6 × 3.15 = 17.64 because b) 3 × 5 = 15 8 11 88 4. a) 53.55 ÷ 6.3 = 8.5 because b) 2 ÷ 1 = 2 × 7 = 14 or 4 2 3 3 3 7 3 1 © SOFAD 0.19 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I 5. a) x + 4 = 9 c) b) 3x – 1 = 8 x=9–4 3x = 8 + 1 x=5 3x = 9 x= 9 3 x=3 4x – 2 = 3 4x = 3 4x = 3 7 9 7 +2 9 25 9 x = 25 × 3 9 4 x = 25 9 d) 2(x + 4) = 17 2x + 8 = 17 2x = 17 – 8 2x = 9 x= 9 2 6. a) Let x be the number of marbles. 17 + x = 23 x = 23 – 17 x=6 Sophia gave Josie 6 marbles. b) Let x be Joey's present salary. 2x = 357.5 x = 357.5 2 x = 178.75 Joey's present salary is $178.75 per week. 0.20 © SOFAD TABLE OF CONTENTS MTH-2007-2 ANSWER KEY Geometry I c) Let x be the width of the window. 3x + 2 = 6.5 3x = 6.5 – 2 3x = 4.5 x = 4.5 3 x = 1.5 The width of the window in Ms Leddy's cottage is 1.5 metres. © SOFAD 0.21 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I ANALYSIS OF THE DIAGNOSTIC TEST RESULTS Question 1. a) b) 2. a) b) 3. a) b) 4. a) b) 5. a) b) c) d) 6. a) b) c) Answer Correct Incorrect Review Section Page Before Going to Unit(s) 10.2 10.2 10.3 10.3 10.4 10.4 10.5 10.5 10.6 10.6 10.6 10.6 10.1 10.1 10.1 10.20 10.20 10.26 10.26 10.32 10.32 10.39 10.39 10.44 10.44 10.44 10.44 10.4 10.4 10.4 7 7 7 7 7 7 7 7 7 7 7 7 1-7-8 1-7-8 1-7-8 • If all your answers are correct, you may begin working on this module. • For each incorrect answer, find the related section listed in the Review column. Do the review activities for that section before beginning the units listed in the right-hand column under the heading Before Going to Unit(s). © SOFAD 0.23 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I INFORMATION FOR EDUCATION STUDENTS DISTANCE You now have the learning material for MTH-2007-2 (GSM-222) together with the homework assignments. Enclosed with this material is a letter of introduction from your tutor indicating the various ways in which you can communicate with him or her (e.g. by letter, telephone) as well as the times when he or she is available. Your tutor will correct your work and help you with your studies. Do not hesitate to make use of his or her services if you have any questions. DEVELOPING EFFECTIVE STUDY HABITS Distance education is a process which offers considerable flexibility, but which also requires active involvement on your part. It demands regular study and sustained effort. Efficient study habits will simplify your task. To ensure effective and continuous progress in your studies, it is strongly recommended that you: • draw up a study timetable that takes your working habits into account and is compatible with your leisure time and other activities; • develop a habit of regular and concentrated study. The following guidelines concerning the theory, examples, exercises and assignments are designed to help you succeed in this mathematics course. © SOFAD 0.25 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I Theory To make sure you thoroughly grasp the theoretical concepts: 1. Read the lesson carefully and underline the important points. 2. Memorize the definitions, formulas and procedures used to solve a given problem, since this will make the lesson much easier to understand. 3. At the end of an assignment, make a note of any points that you do not understand. Your tutor will then be able to give you pertinent explanations. 4. Try to continue studying even if you run into a particular problem. However, if a major difficulty hinders your learning, ask for explanations before sending in your assignment. Contact your tutor, using the procedure outlined in his or her letter of introduction. Examples The examples given throughout the course are an application of the theory you are studying. They illustrate the steps involved in doing the exercises. Carefully study the solutions given in the examples and redo them yourself before starting the exercises. 0.26 © SOFAD TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I Exercises The exercises in each unit are generally modelled on the examples provided. Here are a few suggestions to help you complete these exercises. 1. Write up your solutions, using the examples in the unit as models. It is important not to refer to the answer key found on the coloured pages at the end of the module until you have completed the exercises. 2. Compare your solutions with those in the answer key only after having done all the exercises. Careful! Examine the steps in your solution carefully even if your answers are correct. 3. If you find a mistake in your answer or your solution, review the concepts that you did not understand, as well as the pertinent examples. Then, redo the exercise. 4. Make sure you have successfully completed all the exercises in a unit before moving on to the next one. Homework Assignments Module MTH-2007-2 (GSM-222) contains three assignments. The first page of each assignment indicates the units to which the questions refer. The assignments are designed to evaluate how well you have understood the material studied. They also provide a means of communicating with your tutor. When you have understood the material and have successfully done the pertinent exercises, do the corresponding assignment immediately. Here are a few suggestions. 1. Do a rough draft first and then, if necessary, revise your solutions before submitting a clean copy of your answer. © SOFAD 0.27 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I 2. Copy out your final answers or solutions in the blank spaces of the document to be sent to your tutor. It is preferable to use a pencil. 3. Include a clear and detailed solution with the answer if the problem involves several steps. 4. Mail only one homework assignment at a time. After correcting the assignment, your tutor will return it to you. In the section “Student’s Questions”, write any questions which you may wish to have answered by your tutor. He or she will give you advice and guide you in your studies, if necessary. In this course Homework Assignment 1 is based on Units 1 to 4. Homework Assignment 2 is based on Units 5 to 8. Homework Assignment 3 is based on Units 1 to 8. CERTIFICATION When you have completed all the work, and provided you have maintained an average of at least 60%, you will be eligible to write the examination for this course. 0.28 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I START UNIT 1 BASIC GEOMETRIC CONCEPTS 1.1 SETTING THE CONTEXT Summer holidays in Quebec Alda and Robert live in Montréal. Right now they are on holiday. They have decided to tour their province, the beautiful province of Quebec, but they do not have a specific itinerary in mind. To get an idea of the cities and towns which they could visit, they consult a road map of Quebec that was drawn by one of their friends. Amos Towards James Bay Manicouagan Chibougamau Sept-Îles Saint-Lawrence Baie-Comeau Lake St. Jean Roberval Témiscamingue Matane Chicoutimi Amqui Tadoussac St-Siméon Baie-St-Paul Rivière-du-Loup Québec Lévis Mont-Laurier Trois-Rivières Ontario Matapédia New Brunswick ValléeJonction Thetford-Mines Drummondville St-Georges Hull Sorel Montréal Gaspé Rimouski Lake Mégantic Sherbrooke United States Fig. 1.1 Map of the main highways in Quebec © SOFAD 1.1 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I Like Alda and Robert, look closely at Figure 1.1. What is the connection between this map and geometry? To answer this question, it is important to note the following points: a) highways are generally represented by straight lines; b) many highways leaving the same city can lead to different destinations; c) highways can cross one another; d) some highways extend beyond the edge of the map; e) cities are indicated by black dots. The example provided by this Quebec road map will help you better understand the concepts related to a line. To reach the objective of this unit, you should be able to identify a line, a ray and a line segment in a plane. You should also know how to identify and to measure a right angle, an acute angle, an obtuse angle and a straight angle, using a protractor. Each city is represented by a black dot on the map. In geometry, a point can be thought of as the meeting of two fine pencil marks. The tip of a needle, a grain of sand on a table or even a star in the sky, are "good" representations of a point, but they are inaccurate since a geometric point has no dimensions and only shows position. 1.2 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I A geometric point has neither thickness, width, nor length. Its image must be as small as possible. As on a road map, a geometric point is represented by a small bold dot and is identified by a capital letter. • A Fig. 1.2 • B • C Representation of the points A, B and C on a line Let us return to our vacationers. Alda and Robert are wondering which cities they would like to explore. As they look at the map in Figure 1.1, they consider going to Baie-Comeau since the North Shore is new to them. Once in BaieComeau they could either continue towards Sept-Îles or take the road leading to Manicouagan. They could then visit the Daniel Johnson dam and the various Manicouagan dams. What an interesting trip to look forward to! Look closely at the Montréal–Baie-Comeau–Sept-Îles itinerary shown in Figure 1.1 and answer the following questions: ? What particular feature do you notice about the route going from Montréal to Sept-Îles? ........................................................................................................................... ........................................................................................................................... © SOFAD 1.3 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 ? Geometry I How is the route connecting Baie-Comeau and Manicouagan represented? ........................................................................................................................... ........................................................................................................................... You may have noticed that the route going from Montréal to Sept-Îles extends off the map on both sides. In geometry, this representation of a straight line, with neither beginning nor end, is called a line. A line is unlimited in both directions; it has no beginning or end. Nothing exists in nature which can represent a line in an absolute fashion. But we can imagine a line by looking, for example, at one track of a railroad that runs in a straight line, a very tight telephone wire, or a ray of light. Like a point, a line has neither thickness nor width. It has only length (unlimited). Its image must therefore be drawn as finely as possible with the help of a straightedge. We represent a line as follows: • • A B Fig. 1.3 Representation of the line AB 1.4 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I Notice that a line can be identified by two of its points (written in capital letters). The order in which these two points are named is not important. In Figure 1.3, line AB or line BA are two ways of naming the same line. Now look at the route connecting Baie-Comeau and Manicouagan. This route starts at Baie-Comeau and then extends off the map. In geometry, a straight line which has a beginning but no end is called a ray. A ray is a part of a line. It is unlimited in one direction and limited in the other direction by a point called an endpoint. We represent a ray as follows: • • A B or • • D C Fig. 1.4 Representation of the rays AB and CD Figure 1.4 represents the ray AB, whose endpoint is point A, and the ray CD, which has point C as its endpoint. The first point which is given when naming a ray must be the endpoint of the ray. To return to Figure 1.1, Robert and Alda start off towards Baie-Comeau. Upon arrival in Québec, they decide to make a "brief" detour via Chicoutimi. © SOFAD 1.5 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 ? Geometry I Compared to the two routes you have just studied, what difference do you notice in the route connecting Québec and Chicoutimi? ........................................................................................................................... In fact, the representation of this route shows that it has a beginning (at Québec) and an end (at Chicoutimi). In geometry, a part of a line bounded by two points is called a line segment. A line segment is a part of a line bounded at both ends. We represent a line segment as follows: • • A Fig. 1.5 B Representation of the line segment AB N.B. Line segment AB is identified by the symbol AB. We must always identify a line segment by placing a bar above the two capital letters which name it. (Examples: AB, CD, FG, and so on.) As when naming a line, the order of the letters does not change the figure. There are many examples of objects representing line segments: the edge of a ruler, the intersection of two walls, a step, or a wire joining two nails. It is now time to put these new concepts to use. 1.6 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I Exercise 1.1 1. How many lines can be drawn through a point? ............................................ 2. How many lines can be drawn through two points?....................................... 3. Can the length of a line be measured? Explain your answer. ..................... 4. Can the length of a ray be measured? ............................................................. Explain your answer. ...................................................................................... 5. Can the length of a line segment be measured? ............................................. Explain your answer. ...................................................................................... 6. Look at the figure below and answer the questions that follow. • D • • C A • • F • • B • E G J • H a) Find all the lines which are represented and name them by their points. ....................................................................................................................... b) Identify the points which belong to three different lines. ....................................................................................................................... © SOFAD 1.7 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I c) Identify the points which belong to two different lines. ....................................................................................................................... d) Which points belong to only one line? ....................................................................................................................... e) Identify three rays. ....................................................................................................................... f) Identify ten line segments. Give your answer, using the proper notation for line segments. ....................................................................................................................... 7. Find two examples of objects which represent: a) a line segment ............................................................................................. b) a line ............................................................................................................ c) a ray ............................................................................................................. Let us continue following Alda and Robert on their trip to the North Shore. After having visited Chicoutimi, they head towards Tadoussac. What great scenery where the Saguenay runs into the St Lawrence River! They then continue on towards Baie-Comeau. In Figure 1.6, notice that two routes start at Baie-Comeau. Manicouagan • •Sept-Îles • Baie-Comeau Fig. 1.6 Enlargement of part of Figure 1.1 1.8 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I As seen above, these two routes are represented by rays in geometry. In geometry, the union of two rays form an angle. An angle is a figure formed by the union of two rays with a common endpoint called the vertex. An angle is represented as follows: • A • O A Fig. 1.7 Representation of an angle Point O is the vertex of the angle. Rays OA and OB are the sides of the angle. N.B. An angle is always identified by the symbol ∠. There are three ways of identifying an angle: 1. An angle can be identified by the capital letter representing its vertex whenever there is only one angle originating at this vertex: angle O or ∠O. © SOFAD 1.9 O TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I 2. An angle can be identified using A • three capital letters by placing the letter corresponding to the vertex between the other two: angle AOB • O B or ∠AOB or angle BOA or ∠BOA. 3. An angle can also be identified using a lower-case letter or a number inscribed in the interior of the angle, near the vertex: angle a or ∠a a O and angle 1 or ∠1. and 1 O Look closely at the map in Figure 1.1; you will find several angles whose openings vary in size from one to the other. Look carefully at the following two angles: Ontario Manicouagan • Sherbrooke • • Sept-Îles • Baie-Comeau États-Unis Fig. 1.8 Enlargement of a part of Figure 1.1 1.10 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I Notice that the second angle is larger than the first, because it is more "open". How can the difference in the size of the opening of these two angles be measured? In geometry, several instruments allow us to measure the parts of a figure, or to draw a figure. Here are the definitions and illustrations of the principal geometric instruments which you will use to solve geometric problems: A ruler is an instrument used to measure lengths. It is graduated in centimetres and millimetres. 7 8 9 60 80 70 6 10 11 50 90 5 40 100 4 12 30 110 3 13 14 20 2 0 mm 10 130 120 0 cm 1 140 150 Fig. 1.9 Ruler graduated in cm and mm The set square is an instrument which is used to check and draw right angles. Set square with angles of 90° and 45° Set square with angles of 90°, 60° and 30° Carpenter's set square Industrial designer's T-square Fig. 1.10 Different models of set squares © SOFAD 1.11 15 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I A compass is an instrument which is used to draw circles or arcs of circles. Compass point Drawing point Fig. 1.11 A compass A protractor is an instrument which is used to measure and draw angles. It is graduated in degrees. Edge Zero line Center of the protractor Fig. 1.12 A protractor 1.12 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I A protractor will be used to measure the difference in the size of the openings of the two angles shown previously. The unit of measure for angles is called the degree and its symbol is °. A protractor is divided into degrees from 0° to 180° and its principal parts are the edge, the zero line and the center point. To measure any angle with a protractor: 1. Place the center of the protractor on the vertex of the angle. 2. Place the zero line on one of the sides of the angle; follow the scale which begins at 0° on this side. 3. Read the number of degrees corresponding to the second side of the angle along the edge. (Be careful to read the graduation on the appropriate scale.) A • Example 1 To measure ∠AOB: • O B A • 1. Place the center of the protractor on vertex O of the angle. O © SOFAD 1.13 • B ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I A 2. Place the zero line on • side OB of the angle and follow the scale which starts at 0° close to point B. • O B A 3. Read along the edge • the number of degrees corresponding to side OA of angle AOB. • O B Here m ∠AOB = 60°. ? Using your protractor, find the measures of the two angles in Figure 1.13. a) b) Ontario Sherbrooke • Manicouagan • • Sept-Îles • Baie-Comeau États-Unis Fig. 1.13 Enlargement of a part of Figure 1.1 1.14 © SOFAD TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I To read the measure of an angle on a protractor correctly, it is often useful to extend the sides of the angle a bit using a straightedge, in order to identify precisely to which degree the second side of the angle corresponds. By extending the sides of the angles given in Figure 1.13, we find the measures to be 62° and 97° respectively. N.B. Because of the lack of precision of certain protractors, answers can vary by a few degrees. In all the exercises, a margin of error of 2° is tolerated. After the angles are measured using a protractor, they are classified according to their value in degrees. Here are the definitions of the principal types of angles which you should know. A right angle is an angle which measures 90°. It is the angle which is the most familiar to all of us. We often indicate that an angle is a right angle by placing the symbol A at its vertex. • • B C Fig. 1.14 Right angle ABC Angle ABC is a right angle. It measures 90°. We write m∠ABC = 90°, where the "m" represents the word "measure." © SOFAD 1.15 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I An acute angle is an angle whose measure is less than 90°. The following angles are acute angles: a) b) c) B• •D a A C Fig. 1.15 Representation of acute angles ? Measure the angles in Figure 1.15 using a protractor. a) m∠a = ................... b) m∠BCD = .............. c) m∠A = ....................... They are acute angles since m∠a = 40°, m∠BCD = 60° and m∠A = 45°. A straight angle is an angle which measures 180°. 1.16 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I In a straight angle, the sides of the angle are placed such that one is an extension of the other. The following angles are straight angles: • • A A • • 1 • O B Fig. 1.16 Representation of straight angles In fact, m∠A = 180°, m∠AOB = 180°, m∠1 = 180°; they can also be written in the following way: m∠A = m∠A0B = m∠1 = 180°. An obtuse angle is an angle which measures between 90° and 180°. The angles shown in Figure 1.17 are obtuse angles. b) a) c) •D • C O 3 O Fig. 1.17 Representation of obtuse angles ? Measure the angles in Figure 1.17 using a protractor. a) .............................. © SOFAD b) ............................... 1.17 c) ................................... ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I They are obtuse angles since m∠O = 110°, m∠COD = 120° and m∠3 = 150°. Two other definitions are sometimes used to classify angles: a) an angle less than a straight angle is called a convex angle; b) an angle greater than a straight angle is called a concave angle. In the following figures, angles 1, 2 and 3 are convex angles and angle 4 is a concave angle. 4 2 1 3 Fig. 1.18 Convex and concave angles It is now your turn to apply the concepts you have learned about angles. Exercise 1.2 1. Figure 1.19 represents the layout of a baseball diamond. Which type of angles does it contain? .......................................... 2nd base Pitcher 1st base 3rd base Batter Catcher Fig. 1.19 Layout of a baseball diamond 1.18 © SOFAD ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I 2. a) Using a protractor, measure each of the angles in the figures below (within a margin of error of 2°). b) Qualify each angles as either: right, acute, straight, obtuse or concave. A 2° 1° 1 I 2 6 H 3 B G D 5 C 4 F E a) m∠1 = .......... b) ................ a) m∠A = .......... b) ................ m∠2 = .......... ................ m∠B = .......... ................ m∠3 = .......... ................ m∠C = .......... ................ m∠4 = .......... ................ m∠D = .......... ................ m∠5 = .......... ................ m∠E = .......... ................ m∠6 = .......... ................ m∠F = .......... ................ m∠G = .......... ................ m∠H = ......... ................ m∠I = ......... ................ 3. Complete the following sentences by writing the missing term(s) in the blank spaces. a) Two right angles joined together at their vertex form a .................angle. © SOFAD 1.19 ANSWER KEY TABLE OF CONTENTS MTH-2007-2 Geometry I b) By joining two 35° angles together at their vertex, you obtain an angle called an .......................... angle. c) The angle formed by joining an acute angle and a right angle is called an .......................... angle. d) By joining two 45° angles at their vertex, you obtain an angle called a .......................... angle. e) The angle formed by joining a right angle and a straight angle is called a .......................... angle. 4. Identify the required angles in the figure below. F E A • •C •G H B I • D• J • a) Identify all the acute angles. ....................................................................................................................... b) Identify all the right angles. ....................................................................................................................... c) Identify all the obtuse angles. ....................................................................................................................... d) Identify all the straight angles. ....................................................................................................................... 1.20 © SOFAD TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I Alda and Robert are now on their way back to Montréal, very happy to have visited such a beautiful part of the country. They have every intention of returning. From now on, they will know how to locate the principal cities in Quebec in relation to each other. They now know that Sherbrooke forms an obtuse angle with the route towards Ontario and the route towards the United States, and that Baie-Comeau forms a straight angle with Montréal and SeptÎles. They also know the difference between a line, a ray and a line segment. Check to see whether you too have mastered the concepts covered in this unit. Did you know that the angle which is the most widely known and the most often used is the right angle (90°)? You need only look at the objects around you to be convinced. For example, bricks are placed to form right angles at corners so that they can be stacked properly. Masons (or bricklayers) make themselves set squares with pieces of cord, one placed horizontally with the help of a level and the other placed vertically with the help of a plumb line. They use this improvised set square as a reference point to align the bricks correctly. In Ancient Egypt, builders and surveyors made a type of set square with a rope separated into 12 equal parts by 11 knots. One assistant held the two ends of the rope together, another held the third knot from one of the ends and a third assistant held the fifth knot from the other end. Pulled in this way, the rope formed a right angle which served as a set square for the builders. © SOFAD 1.21 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I 5 4 3 Fig. 1.20 Rope with 11 knots used as a set square These Egyptians were creative, were they not! 1.22 © SOFAD TABLE OF CONTENTS ANSWER KEY MTH-2007-2 ? 1.2 Geometry I PRACTICE EXERCISES 1. In the adjacent figure, find the number of points determined by the intersection of two line segments. Answer: ............................................ 2. Identify the geometric representation of the adjacent figure: A B • Answer: ............................................ • 3. By taking point A as the endpoint, draw 3 rays. •A 4. Study the figure below. B • c 1 • 2 A • D • F C f 3 • H 4 i • I © SOFAD G •E 1.23 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I a) Identify the vertex of each of the following angles by the corresponding letter: 1. ∠ACD .............. 2. ∠2 ............... 4. ∠3 ............... 5. ∠f ............... 3. ∠I ............... b) Using a protractor, measure the following angles in degrees: 1. ∠ACB .............. 2. ∠GFH ............... 4. ∠c ............... 5. ∠ECF ............... 3. ∠4 ............... (A margin of error of 2° is acceptable.) c) Identify each of the following, using three letters: 1. all the acute angles ................................................................................................................ 2. all the obtuse angles ................................................................................................................ 3. all the straight angles ................................................................................................................ 4. all the right angles ................................................................................................................ 5. When a clock says 4 o'clock, the angle formed by its hands measures ..................... degrees. 1.24 © SOFAD TABLE OF CONTENTS ANSWER KEY MTH-2007-2 1.3 Geometry I SUMMARY ACTIVITY Answer the following questions. Reread Unit 1 if necessary. 1. What is a ray? ........................................................................................................................... ........................................................................................................................... 2. What is a line segment? ........................................................................................................................... ........................................................................................................................... 3. What is the difference between a line and a ray? ........................................................................................................................... ........................................................................................................................... 4. Complete the following sentences by writing in the missing term(s) in the blank spaces. a) An angle is a figure formed by the union of two ....................... originating from the same point called the ..................... . b) A protractor is an instrument used to ............................................... and .................... an angle. c) An ............................. angle is an angle whose measure is less than 90°. © SOFAD 1.25 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I d) An .................... angle is an angle which measures between 90° and 180°. e) An angle which measures ................. is a straight angle. f) An angle which measures ................. is a right angle. 5. Using a protractor, measure angle PRS in the adjacent figure and describe the steps which you take to obtain this measure. P • R • S m∠PRS = ......................... 1. ..................................................................................................................... ..................................................................................................................... 2. ..................................................................................................................... ..................................................................................................................... 3. ..................................................................................................................... ..................................................................................................................... 6. Define the following symbols by completing the sentences: a) AB signifies ................................................................................................. b) ∠A signifies ................................................................................................. c) m∠AOB signifies ......................................................................................... d) 30° signifies 30 ............................................................................................ 1.26 © SOFAD TABLE OF CONTENTS ANSWER KEY MTH-2007-2 1.4 Geometry I THE MATHWHIZ PAGE Measuring angles It is often necessary to know how to measure angles in everyday life. What type of people might need to know the measures of different angles? What instruments would they use? The answers to these two questions are complex. A protractor is used in geometry to measure angles, but it is not the only instrument used for this purpose. In fact, ship's captains must be able to determine their ship's position on a marine map in order to follow the designated route. To do this, they must be able to determine their longitude and latitude. To do so, they measure the angle between the horizon and the sun, moon or North Star. They use a sextant to measure this angle. © SOFAD 1.27 TABLE OF CONTENTS ANSWER KEY MTH-2007-2 Geometry I Fig. 1.21 The sextant When they know the measure of the angle, they use a chronometer to find out the time at Greenwich, England and study astronomical charts to find out the attitudes of celestial bodies at Greenwich for any time and any day. The configuration of these bodies in the sky is different. Compared to information on the astronomical charts, this configuration seems to be turned at an angle. By recording these two pieces of information on a nautical map, ship's captains can determine their exact location. 1.28 © SOFAD