MTH-2007 C1 Layout 1 - SOFAD · GSM-222-01 Basic Geometric Concepts To distinguish between the...

Geometry IMTH-2007-2

GEOMETRY I

MTH-2007-2

This course was produced in collaboration with the Service de l'éducationdes adultes de la Commission scolaire catholique de Sherbrooke and theState Secretary of Canada.

Author: Marie-Reine Rouillard

Content revision: Jean-Paul GroleauDiane 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. Anyreproduction, by mechanical or electronic means, including micro-reproduction, is forbidden withoutthe written permission of a duly authorized representative of the Société de formation à distance descommissions scolaires du Québec (SOFAD).

Legal Deposit – 2001

Bibliothèque et Archives nationales du Québec

Bibliothèque et Archives Canada

ISBN 978-2-89493-198-1

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MTH-2007-2 Geometry I

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TABLE OF CONTENTS

Introduction to the Program Flowchart ................................................... 0.4The Program Flowchart ............................................................................ 0.5How to Use this Guide .............................................................................. 0.6General Introduction................................................................................. 0.9Intermediate and Terminal Objectives of the Module ............................ 0.10Diagnostic Test on the Prerequisites ....................................................... 0.15Answer Key for the Diagnostic Test on the Prerequisites ...................... 0.19Analysis of the Diagnostic Test Results ................................................... 0.23Information for Distance Education Students......................................... 0.25

UNITS

1. Basic Geometric Concepts......................................................................... 1.12. Drawing an Angle ..................................................................................... 2.13. Types of Lines ............................................................................................ 3.14. Categories of Angles .................................................................................. 4.15. Polygons ..................................................................................................... 5.16. Measuring Polygons .................................................................................. 6.17. Pythagoras' Theorem ................................................................................ 7.18. Special Right Triangles and Pythagoras' Theorem ................................. 8.1

Final Review .............................................................................................. 9.1Terminal Objectives .................................................................................. 9.5Self-Evaluation Test.................................................................................. 9.7Answer Key for the Self-Evaluation Test ................................................ 9.13Analysis of the Self-Evaluation Test Results .......................................... 9.17Final Evaluation........................................................................................ 9.18Answer Key for the Exercises ................................................................... 9.19Glossary ..................................................................................................... 9.71List of Symbols .......................................................................................... 9.76Bibliography .............................................................................................. 9.77

Review Activities ..................................................................................... 10.1

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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!

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THE PROGRAM FLOWCHART

CEGEP

MTH-5110-1 Introduction to Vectors

MTH-5109-1 Geometry IV

MTH-5108-2 Trigonometric Functions and Equations

MTH-5107-2 Exponential and Logarithmic Functions and Equations

MTH-5106-1 Real Functions and Equations

MTH-5105-1 Conics

MTH-5104-1 Optimization II

MTH-5103-1 Probability II

MTH-5102-1 Statistics III

MTH-5101-1 Optimization I

MTH-4110-1 The Four Operations on Algebraic Fractions

MTH-4109-1 Sets, Relations and Functions

MTH-4108-1 Quadratic Functions

MTH-4107-1 Straight Lines II

MTH-4106-1 Factoring and Algebraic Functions

MTH-4105-1 Exponents and Radicals

MTH-4103-1 Trigonometry I

MTH-4102-1 Geometry III

MTH-536

MTH-526

MTH-514

MTH-436

MTH-426

MTH-416

MTH-314

MTH-216

MTH-116

MTH-3002-2 Geometry II

MTH-3001-2 The Four Operations on Polynomials

MTH-2008-2 Statistics and Probabilities I


MTH-2006-2 Equations and Inequalities I

MTH-1007-2 Decimals and Percent

MTH-1006-2 The Four Operations on Fractions

MTH-1005-2 The Four Operations on Integers

MTH-5111-2 Complement and Synthesis II

MTH-4111-2 Complement and Synthesis I

MTH-4101-2 Equations and Inequalities II

MTH-3003-2 Straight Lines I

TradesDVS

MTH-5112-1 Logic

25 hours = 1 credit

50 hours = 2 credits

MTH-4104-2 Statistics II

You ar e here

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Hi! My name is Monica and I have beenasked to tell you about this math module.What’s your name?

I’m Andy.

Whether you areregistered at anadult education

center or atFormation àdistance, ...

You’ll see that with this method, math isa real breeze!

... you have probably taken aplacement test which tells youexactly which module youshould start with.

My results on the testindicate that I should beginwith this module.

Now, the module you have in yourhand is divided into threesections. The first section is...

... the entry activity, whichcontains the test on theprerequisites.

By carefully correcting this test using thecorresponding answer key, and record-ing your results on the analysis sheet ...

HOW TO USE THIS GUIDE

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?

The memo pad signals a brief reminder ofconcepts which you have already studied.

The calculator symbol reminds you thatyou will need to use your calculator.

The sheaf of wheat indicates a review designed toreinforce what you have just learned. A row ofsheaves near the end of the module indicates thefinal review, which helps you to interrelate all thelearning activities in the module.

The starting lineshows where thelearning activitiesbegin.

The little white question mark indicates the questionsfor which answers are given in the text.?

... you can tell if you’re well enoughprepared to do all the activities in themodule.

The boldface question markindicates practice exerciceswhich allow you to try out whatyou have just learned.

And if I’m not, if I need a littlereview before moving on, whathappens then?

In that case, before you start theactivities in the module, the resultsanalysis chart refers you to a reviewactivity near the end of the module.

In this way, I can be sure Ihave all the prerequisitesfor starting.

Exactly! The second sectioncontains the learning activities. It’sthe main part of the module.

Look closely at the box tothe right. It explains thesymbols used to identify thevarious activities.

The target precedes theobjective to be met.

I see!

?

START

Lastly, the finish line indicatesthat it is time to go on to the self-evaluationtest to verify how well you have understoodthe learning activities.

FINISH

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A “Did you know that...”?

Later ...

For example. words in bold-face italics appear in theglossary at the end of themodule...

G r e a t !

... statements in boxes are importantpoints to remember, like definitions, for-mulas and rules. I’m telling you, theformat makes everything much easier.

The third section contains the final re-view, which interrelates the differentparts of the module.

Yes, for example, short tidbitson the history of mathematicsand fun puzzles. They are in-teresting and relieve tension atthe same time.

No, it’s not part of the learn-ing activity. It’s just there togive you a breather.

There are also many fun thingsin this module. For example,when you see the drawing of asage, it introduces a “Did youknow that...”

Must I memorize what the sage says?

It’s the same for the “math whiz”pages, which are designed espe-cially for those who love math.

They are so stimulating thateven if you don’t have to dothem, you’ll still want to.

And the whole module hasbeen arranged to makelearning easier.

There is also a self-evaluationtest and answer key. They tellyou if you’re ready for the finalevaluation.

Thanks, Monica, you’ve been a bighelp.

I’m glad! Now,I’ve got to run.

See you!This is great! I never thought that I wouldlike mathematics as much as this!

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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 attri-buted to an Egyptian pharaoh who hired "land measurers" to re-establish landboundaries along the shores of the Nile after the spring floods. Since the fencesand boundary markers were destroyed yearly by the rising waters, the pharaohhad the lands measured each year so that he could readjust the taxes.

It is thought that Nature probably provided the inspiration for a number ofgeometric 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 providesexamples of most of the geometric elements or figures studied in this module: thepoint, the line, the ray, the line segment, the right, acute, straight or obtuseangle, 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 appropriateinstruments on hand. Your geometry set should include: a ruler graduated incentimetres and millimetres, a right triangle, a protractor and acompass. A description of these instruments is given in the first unit.

The concept of line is covered in detail to enable you to distinguish intersec-ting lines (such as street intersections: X), perpendicular lines (such as antennain 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 andtheir characteristics. In it you will learn about complementary andsupplementary angles, adjacent angles, vertically opposite angles, alternateexterior and alternate interior angles and finally, corresponding angles.

The last activity involves studying Pythagoras' theorem and its applicationsto various right triangles.

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INTERMEDIATE AND TERMINAL OBJECTIVES OFTHE 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)

GSM-222-01

and 10 20%

GSM-222-02

GSM-222-03

and 12 20%

GSM-222-04

GSM-222-05

and 13 30%

GSM-222-06

GSM-222-07 8 20%

GSM-222-08 5 10%

* GSM stands for “General Education, Secondary-Level, Mathematics.”

** Two hours are allotted for the final evaluation.

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GSM-222-01 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.

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GSM 222-05 Polygons

Given a set of geometric figures illustrating polygons, to re-

cognize 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.

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GSM 222-08 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.

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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 diagnos-

tic 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.

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1. Perform the following additions.

a) 1054.6 + 956.31 = .............................. because 1054.6

b) 37 + 1

2 =.......................................................................................................

2. Perform the following subtractions.

a) 861 – 456.3 = ..................................... because 861

–

b) 59 – 1

4 =.......................................................................................................

3. Perform the following multiplications.

a) 5.6 × 3.15 = ........................................ because 3.15

×

b) 38 × 5

11 =.....................................................................................................

4. Perform the following divisions.

a) 53.55 ÷ 6.3 = ...................................... because 5355

b) 23 ÷ 1

7 =.......................................................................................................

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5. Solve each of the equations below. The complete solution is required.

a) x + 4 = 9 b) 3x – 1 = 8

c) 4x3 – 2 = 7

9 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?

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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?

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ANSWER KEY FOR THE DIAGNOSTIC TESTON THE PREREQUISITES

1. a) 1 054.6 + 956.31 = 2 010.91 because

b) 37 + 1

2 = 614 + 7

14 = 1314

2. a) 861 – 456.3 = 404.7 because

b) 59 – 1

4 = 2036 – 9

36 = 1136

3. a) 5.6 × 3.15 = 17.64 because

b) 38 × 5

11 = 1588

4. a) 53.55 ÷ 6.3 = 8.5 because

b) 23 ÷ 1

7 = 23 × 7

1 = 143 or 4 2

3

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5. a) x + 4 = 9 b) 3x – 1 = 8

x = 9 – 4 3x = 8 + 1

x = 5 3x = 9

x = 93

x = 3

c) 4x3 – 2 = 7

9 d) 2(x + 4) = 17

4x3 = 7

9 + 2 2x + 8 = 17

4x3 = 25

9 2x = 17 – 8

x = 259 × 3

4 2x = 9

x = 259 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 = 357.52

x = 178.75

Joey's present salary is $178.75 per week.

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c) Let x be the width of the window.

3x + 2 = 6.5

3x = 6.5 – 2

3x = 4.5

x = 4.53

x = 1.5

The width of the window in Ms Leddy's cottage is 1.5 metres.

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ANALYSIS OF THE DIAGNOSTICTEST RESULTS

QuestionAnswer Review Before Going

Correct Incorrect Section Page to Unit(s)

1. a) 10.2 10.20 7b) 10.2 10.20 7

2. a) 10.3 10.26 7b) 10.3 10.26 7

3. a) 10.4 10.32 7b) 10.4 10.32 7

4. a) 10.5 10.39 7b) 10.5 10.39 7

5. a) 10.6 10.44 7b) 10.6 10.44 7c) 10.6 10.44 7d) 10.6 10.44 7

6. a) 10.1 10.4 1-7-8b) 10.1 10.4 1-7-8c) 10.1 10.4 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).

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INFORMATION FOR DISTANCEEDUCATION STUDENTS

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.

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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.

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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.

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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 assign-

ment, 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.



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.

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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.

Roberval

Amos

Ontario

Témiscamingue

Manicouagan

United States

Lake Mégantic

Baie-Comeau

Sept-ÎlesChibougamau

Mont-Laurier

Hull

Montréal

New Brunswick

Rivière-du-Loup

Rimouski

Matane

Matapédia

Towards James Bay

Trois-Rivières

Québec

Baie-St-Paul

St-Georges

Chicoutimi

Tadoussac

DrummondvilleThetford-Mines

Lévis

ValléeJonction

St-Siméon

Sorel

Sherbrooke

Gaspé

Saint-Lawrence

Amqui

LakeSt. Jean

Fig. 1.1 Map of the main highways in Quebec

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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 under-

stand 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.

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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.

• •C

•A B

Fig. 1.2 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 Baie-

Comeau 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?

...........................................................................................................................

...........................................................................................................................

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? 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:

• •BA

Fig. 1.3 Representation of the line AB

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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

•D

•C

or

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.

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? 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:

•B

•A

Fig. 1.5 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.

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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

•F

•E•

J

•H

•G

•B

•A

a) Find all the lines which are represented and name them by their points.

.......................................................................................................................

b) Identify the points which belong to three different lines.

.......................................................................................................................

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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

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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

•A

O

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 ver-

tex whenever there is only one angle

originating at this vertex: angle O or

∠O.

O

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2. An angle can be identified using

three capital letters by placing the

letter corresponding to the vertex

between the other two: angle AOB

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

and angle 1 or ∠1.

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:

•

••Manicouagan

Sept-Îles

Baie-Comeau

Ontario Sherbrooke

États-Unis

•

Fig. 1.8 Enlargement of a part of Figure 1.1

•

•A

BO

and

O

O

a

1

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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.

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 Set square with Carpenter's set Industrial de-angles of 90° and angles of 90°, 60° square signer's T-square

45° and 30°

Fig. 1.10 Different models of set squares

0 cm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

100 0 mm102030405060708090110120130140150

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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.

Center of the protractor Zero line

Edge

Fig. 1.12 A protractor

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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.)

Example 1

To measure ∠AOB:

1. Place the center of the

protractor on vertex O

of the angle.

•A

•BO

•A

•BO

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2. Place the zero line on

side OB of the angle

and follow the scale

which starts at 0° close

to point B.

3. Read along the edge

the number of degrees

corresponding to side

OA of angle AOB.

Here m ∠AOB = 60°.

? Using your protractor, find the measures of the two angles in Figure 1.13.

a)

•

••Manicouagan

Sept-Îles

Baie-Comeau

b)Ontario Sherbrooke

États-Unis

•

Fig. 1.13 Enlargement of a part of Figure 1.1

•A

•BO

•A

•BO

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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 accor-

ding 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 at its vertex.

•

•A

BC

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."

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An acute angle is an angle whose measure is less than 90°.

The following angles are acute angles:

a)

a

b)

••

C

DB

c)

A

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°.

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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:

••

•O

•A

•A

B

1

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.

a)

O

b)

•

•C O

D

c)

3

Fig. 1.17 Representation of obtuse angles

? Measure the angles in Figure 1.17 using a protractor.

a) .............................. b) ............................... c) ...................................

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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.

1

2

3

4

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? ..........................................

Catcher

Batter

1st base

2nd base

3rd base

Pitcher

Fig. 1.19 Layout of a baseball diamond

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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.

1°

1

2

3

45

6

2°

G

I

H

A

B

C

D

EF

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.

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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.

•

•

• •

•

•A

CB

D

H

J

I

G

F

E

a) Identify all the acute angles.

.......................................................................................................................

b) Identify all the right angles.

.......................................................................................................................

c) Identify all the obtuse angles.

.......................................................................................................................

d) Identify all the straight angles.

.......................................................................................................................

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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 Sept-

Îles. 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.

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1.2 PRACTICE EXERCISES

1. In the adjacent figure, find the num-

ber of points determined by the inter-

section of two line segments.

Answer: ............................................

2. Identify the geometric representa-

tion of the adjacent figure:

Answer: ............................................

3. By taking point A as the endpoint, draw 3 rays.

•A

4. Study the figure below.

•

•

•

•

• •

•1

2 3C

c

D

A

B

E

I

i

G

F f

4H

?

• •A B

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a) Identify the vertex of each of the following angles by the corresponding

letter:

1. ∠ACD .............. 2. ∠2 ............... 3. ∠I ...............

4. ∠3 ............... 5. ∠f ...............

b) Using a protractor, measure the following angles in degrees:

1. ∠ACB .............. 2. ∠GFH ............... 3. ∠4 ...............

4. ∠c ............... 5. ∠ECF ...............

(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.

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1.3 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°.

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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 des-

cribe the steps which you take to

obtain this measure.

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 ............................................................................................

•

•

P

R S

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1.4 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.

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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.

MTH-2007 C1 Layout 1 - SOFAD · GSM-222-01 Basic Geometric Concepts To distinguish between the...

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