Version 01 Angles Apprenticeship and Workplace Mathematics (Grade 10/Literacy Foundations Level 7) © 2012 by Open School BC http://mirrors.org/presskit/buttons/88x31/eps/by-nc.eps This work is licensed under the Creative Commons Attribution-NonCommercial 4. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/4.0/ Permissions beyond the scope of this license are identified in the materials by a copyright symbol and are outlined below. To request permission to use the exclusions to this Creative Commons license, contact the author/publisher of the third party materials: Third party copyright exclusions include: All photographs used under license from Shutterstock. The Data Pages were reproduced with permission from the BC Ministry of Education.
Course History New, March 2012 Project Partners This course was developed in partnership with the Distributed Learning Resources Branch of Alberta Education and the following organizations: • Black Gold Regional Schools • Calgary Board of Education • Edmonton Public Schools • Peace Wapiti School Division No. 76 • Pembina Hills Regional Division No. 7 • Rocky View School Division No. 41 Project Management: Jennifer Riddel Content Revisions: Jennifer Riddel, Ester Moreno Edit: Leanne Baugh, Monique Brewer Math Edit: Learning Centre of the Greater Victoria School District Continuing Education Program: Nigel Cocking, Keith Myles, Bill Scott School District 47, Powell River: Tania Hobson OSBC: Christina Teskey Module Tests: Barb Lajeunesse, Michael Finnigan (SD 34) Copyright: Ilona Ugro Production Technicians: Sharon Barker, Beverly Carstensen, Dennis Evans, Brian Glover Art Coordination: Christine Ramkeesoon Media Coordination: Janet Bartz Art: Cal Jones Flash Programming: Sean Cunniam Narration Recording: MOH Productions and Neil Osborne Voice Talent: Felix LeBlanc, Kate Eldridge, Wendy Webb and MOH Productions Advisors: JD Caudle (Yukon Territory), Randy Decker (SD 40), Bev Fairful (Yukon Territory), Sonya Fern (SD 62), Sandra Garfinkel (SD 39), Richard Giroday (SD 58), Sharon Hann (SD 39), Tim Huttemann (SD 20), Dan Laidlaw (SD 73), Heather Lessard (SD 53), Gloria Lowe (SD 6), Jan Malcolm (SD 36), Christina Teskey (OSBC), Jennifer Waughtal (SD 57), Ray Wong (SD 91) Table of Contents Table of Contents Section Organization.
1 Lesson A: Sketching and Measuring Angles. 3 Lesson B: Constructing Congruent Angles. 29 Lesson C: Bisecting Angles. 53 Lesson D: Relationships Among Angles.
75 Lesson E: Parallel and Perpendicular Lines. 101 Lesson F: Solving Problems Using Angles Relationships. 181 © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | iii Viewing Your PDF Learning Package This PDF Learning Package is designed to be viewed in Acrobat. If you are using the optional media resources, you should be able to link directly to the resource from the pdf viewed in Acrobat Reader.
The links may not work as expected with other pdf viewers. Download Adobe Acrobat Reader: http://get.com/reader/ Section Organization Section Organization This section on Angles is made up of several lessons. Lessons Lessons have a combination of reading and hands-on activities to give you a chance to process the material while being an active learner. Each lesson is made up of the following parts: Essential Questions The essential questions included here are based on the main concepts in each lesson.
These help you focus on what you will learn in the lesson. Focus This is a brief introduction to the lesson. Get Started This is a quick refresher of the key information and skills you will need to be successful in the lesson. Activities Throughout the lesson you will see three types of activities: • Try This activities are hands-on, exploratory activities.
• Self-Check activities provide practice with the skills and concepts recently taught. • Mastering Concepts activities extend and apply the skills you learned in the lesson. You will mark these activities using the solutions at the end of each section. Explore Here you will explore new concepts, make predictions, and discover patterns.
Bringing Ideas Together This is the main teaching part of the lesson. Here, you will build on the ideas from the Get Started and the Explore. You will expand your knowledge and practice your new skills. Lesson Summary This is a brief summary of the lesson content as well as some instructions on what to do next.
© OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | v Section Organization At the end of each section you will find: Solutions This contains all of the solutions to the Activities. Appendix Here you will find the Data Pages along with other extra resources that you need to complete the section. You will be directed to these as needed. Glossary This is a list of key terms and their definitions.
Throughout the section, you will see the following features: Icons Throughout the section you will see a few icons used on the left-hand side of the page. These icons are used to signal a change in activity or to bring your attention to important instructions. AWM online resource (optional) This indicates a resource available on the internet. If you do not have access, you may skip these sections.
Solutions vi | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Section Organization My Notes The column on the outside edge of most pages is called “My Notes”. You can use this space to: • write questions about things you don’t understand. • note things that you want to look at again. • draw pictures that help you understand the math.
• identify words that you don’t understand. • connect what you are learning to what you already know. • make your own notes or comments. Materials and Resources There is no textbook required for this course.
You will be expected to have certain tools and materials at your disposal while working on the lessons. When you begin a lesson, have a look at the list of items you will need. You can find this list on the first page of the lesson, right under the lesson title. In general, you should have the following things handy while you work on your lessons: • a scientific calculator • a ruler • a geometry set • Data Pages (found in the Appendix) © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | vii Angles Angles Photo by Larry St.
Pierre © 2010 The game of lacrosse originated among the First Peoples of North America and has been an important part of their culture for almost a thousand years. Recognized by an act of Parliament in 1994 as Canada’s national summer sport, it is a vigorous game enjoyed by young men and women across our country! The graphic is of a contemporary lacrosse stick. The webbing geometric design changes from the pocket to the shooting string to make it easier to either cradle or shoot the ball. The parallel lines, four-sided figures, and the repeated angles in the webbing illustrate that geometry plays an important role in sports equipment.
In this section you will explore the geometry of angles. In particular, you will investigate how they are defined, measured, classified, duplicated, and bisected. As well, you will explore relationships among the angles formed when two parallel lines are cut by a third line. In each case, you will apply definitions and relationships to solve a variety of practical problems.
© OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 1 Angles In this section you will: • classify angles and pairs of angles. • draw, bisect, replicate, and construct a variety of angles. • solve problems that involve parallel, perpendicular and transversal lines, and the pairs of angles formed between them. 2 | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Angles—Lesson A: Sketching and Measuring Angles Lesson A Sketching and Measuring Angles To complete this lesson, you will need: In this lesson, you will complete: • a protractor • 5 activities • a ruler • scissors • a compass • several blank sheets of paper Essential Questions • How can you sketch and describe angles of various measures? • How are referents used to estimate the measure of a given angle? • How is the protractor used to measure angles in a variety of orientations? © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 3 Angles—Lesson A: Sketching and Measuring Angles My Notes Focus Photo by Vitaly M © 2010 Cue sports such as snooker require a steady hand, practice, and an eye for angles.
What separates the novice and the skilled player is the ability to predict how the billiard balls will behave when struck and when they carom of the cushions. Spaced at regular intervals along the table’s perimeter are diamond shapes, which assist a player in assessing possible angles to line up the shot. When a cue ball strikes another, what is the largest possible angle the ball struck can be deflected from the line along which the cue ball was travelling? Get Started To move through this lesson and further through the section you will need to be able to measure and draw angles. In this part of the lesson you will review the parts of an angle and how to sketch and measure angles using a protractor.
4 | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Angles—Lesson A: Sketching and Measuring Angles Parts of an Angle My Notes In previous math courses you drew and measured angles of various sizes. An angle is a geometric shape formed by two rays with a common endpoint. Each ray is called an arm of the angle. The common endpoint of the arms of the angle is the vertex of the angle.
Arm Vertex Measure of an angle Arm The measure of an angle is commonly given in degrees. There are 360 degrees (360º) in one rotation. To see all the angles possible in one 360 degree rotation, go and look at Angle Definition (http://media.ca/ osbcmedia/math/mathawm10/glossary/Division03/Angle/index. Did You Know? In ancient times, the peoples of the Middle East and India divided the circle into 360 parts.
This was partly because there are approximately 360 days in a year and the stars move in the sky in a great circle that takes a year to complete. They didn’t use 365 because the number 360 is much more convenient to use. Besides 1 and 360, the number 360 can be divided evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180. Arithmetic is simpler using 360 than it is using 365! © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 5 Angles—Lesson A: Sketching and Measuring Angles Measuring and Drawing Angles My Notes First review how to use your protractor.
Inner scale Outer scale Centre point The two scales on your protractor are used to measure angles drawn in different orientations. Remember, line up the centre-point of the protractor with the angle’s vertex. Then start at zero on one of the protractor’s scales. Both of the angles pictured below measure 30º.
Use outer Use inner scale scale Obtuse angles can be measured the same way. Both of the angles pictured below measure 120°. Use outer Use inner scale scale Go an look at Protractor (http://media.ca/ osbcmedia/math/mathawm10/glossary/Division03/Protractor/ index. Use the demonstration applet to practise measuring angles.
Watch the angle measurements changing as you drag the coloured dot. 6 | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Angles—Lesson A: Sketching and Measuring Angles Activity 1 My Notes Self-Check To complete this activity, you’ll need your protractor and straightedge. Use a protractor to measure each angle. Record all measurements to the nearest degree.
Draw each angle from the measurements below. Draw the angles in parts a and b opening to the right as in 1. Draw the angles in parts c and d opening to the left as in 1. 49º © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 7 Angles—Lesson A: Sketching and Measuring Angles b.
127º Turn to the solutions at the end of the section and mark your work.