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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
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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
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ANSWER
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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
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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 ...
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ANSWER
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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.
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ANSWER
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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!
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ANSWER
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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.
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TABLE OF
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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.
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TABLE OF
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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.
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TABLE OF
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ANSWER
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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
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TABLE OF
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ANSWER
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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.
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TABLE OF
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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.
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TABLE OF
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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
.......................................................................................................
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TABLE OF
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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?
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TABLE OF
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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?
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TABLE OF
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ANSWER
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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
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ANSWER
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TABLE OF
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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.
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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.
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ANSWER
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TABLE OF
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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).
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TABLE OF
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ANSWER
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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.
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TABLE OF
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ANSWER
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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.
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ANSWER
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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.
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TABLE OF
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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.
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TABLE OF
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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
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1.1
ANSWER
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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
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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?
...........................................................................................................................
...........................................................................................................................
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?
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
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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.
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?
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
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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.
.......................................................................................................................
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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
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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.
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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
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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
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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
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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
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1.13
•
B
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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
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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."
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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
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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) ..............................
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b) ...............................
1.17
c) ...................................
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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
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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.
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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
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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.
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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
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?
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
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G
•E
1.23
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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
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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°.
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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
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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.
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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
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