An Angle on Geometry

For Ages 10+ r o e t s Bo r e

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An Angle on © ReadyEdPubl i cat i ons Geometry •f orr evi ew pur posesonl y• te

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An introduction to geometry, angles, triangles, circles and . other 2D shapes.

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Written by Jane Bourke. Illustrated by Melinda Parker. © Ready-Ed Publications - 1999 Published by Ready-Ed Publications (1999) P.O. Box 276 Greenwood WA 6024 Email: info@readyed.com.au Website: www.readyed.com.au

COPYRIGHT NOTICE Permission is granted for the purchaser to photocopy sufficient copies for non-commercial educational purposes. However this permission is not transferable and applies only to the purchasing individual or institution.

ISBN 1 86397 227 7

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Teachers’ Notes .................................................................... 4 Looking at Different Angles ............................................... 5 Measuring Angles 1 .............................................................. 6 Reflex Angles ......................................................................... 7 Angles on the Line ................................................................ 8 Which Angle is Larger? ........................................................ 9 Naming Angles .................................................................... 10 Measuring Angles 2 ............................................................ 11 Angles in a Triangle ............................................................ 12 Angle Facts .......................................................................... 13 Scalene Triangles ................................................................ 14 Isosceles Triangles .............................................................. 15 Equilateral Triangles .......................................................... 16 Get the Right Angle ........................................................... 17 Angling Around ................................................................... 18 Intersecting Lines ................................................................ 19 Parallel Lines ....................................................................... 20 Degrees in a Circle .............................................................. 21 Constructing Angles 1 ....................................................... 22 Constructing Angles 2 ....................................................... 23 Angles and Directions 1 .................................................... 24 Angles and Directions 2 .................................................... 25 Snooker Angles ................................................................... 26 Angle Check Point .............................................................. 27 Angles of the Game ........................................................... 28 Baseball Hits........................................................................ 29 An Angle on Time .............................................................. 30 Angles in the Real World ................................................... 31 Puzzles With Angles........................................................... 32 Parts of a Circle 1 ............................................................... 33 Parts of a Circle 2 ............................................................... 34 Triangles in Circles 1 .......................................................... 35 Triangles in Circles 2 .......................................................... 36 Angles in Circles 1 .............................................................. 37 Angles in Circles 2 .............................................................. 38 Shapes in Circles 1 .............................................................. 39 Shapes in Circles 2 .............................................................. 40 Answers ................................................................................ 42

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CONTENTS

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TEACHERS’ NOTES This book is designed to complement the geometry component of the space maths strand of the curriculum. It provides a basic introduction to new concepts as well as activities that will consolidate the skills and ideas associated with introductory geometry.

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The book is designed to be used sequentially as certain skills need to be mastered in order to complete some of the later activities. Many of the activity pages explain the various mathematical concepts and provide examples, however, it is assumed that these ideas will be discussed in class prior to students completing the worksheets.

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The activities in this book cover the major learning areas such as identifying different types of angles, using a protractor to measure angles, using known rules to calculate the size of angles and constructing angles using either a compass or a protractor.

Angles in a wide range of 2D objects are explored, specifically, the angles of scalene, isosceles and equilateral triangles, parallel and intersecting lines and angles in a circle. In addition, there are several pages that apply many of these concepts to angles in everyday situations.

The book also explores the mathematics of circles examining features such as chords, arcs, angles and various shapes in circles.

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Additional materials: Before starting this unit of work, ensure that each student has access to a compass, a protractor and a ruler. It is probably best to use pencils rather than pens for construction activities.

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Important notes about diagrams: Occasionally some angles may not appear to be what the answers specify. This is due to slight variations in the printing process and, unfortunately, these differences are beyond our control.

Rays in diagrams would normally have arrow-heads but they have been omitted in this book to allow more room.

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Also, many 90° angles have not been marked with squares to allow diagrams to be more clear.

Angles that look 90° generally are 90° such as those on pages 7, 14 and 32.

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LOOKING AT DIFFERENT ANGLES An angle is the amount of turn between two lines around a common point.

The lines are known as rays and the point at which they meet is called a vertex.

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An acute angle is an angle less than 90°. Draw two more examples below.

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A right angle is an angle that measures exactly 90° They are often marked with a square at the angle.

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An obtuse angle measures between 90° and 180°. Draw two more examples below.

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R Tick the angles below that are right angles. Draw a circle around the acute angles and put a cross inside the angles that are obtuse.

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MEASURING ANGLES 1 Angles are measured in degrees. This is usually expressed with this symbol °. A protractor is used to measure angles.

Using a protractor follow the example below and then complete the activities.

0 0

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r o e t s Bo r e p ok u S To measure an angle:

1. Place the centre of the protractor on the corner or sharpest point (vertex) of the angle.

2. Turn the protractor so that the base line runs along one of the lines that forms the angle.

3. You can then read the size of the angle from the position of the second line. For example this angle is approximately _______° Most protractors number the angles both clockwise and anti-clockwise.

Make sure that you start at 0 and follow the correct set of numbers.

© ReadyEdPubl i cat i ons •angles f obelow rr e i e wthep u r po es l y •or R Measure the andv write down type of angle fors each one,o e.g.n acute, obtuse right.

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size: ...................... type: .....................

d. size: ...................... type: .....................

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size: ......................

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type: .....................

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size: ...................... type: .....................

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size: ......................

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type: .....................

type: .....................

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REFLEX ANGLES A reflex angle is an angle between 180° and 360°. The reflex angle below measures 320°

R 1. Without using a protractor find the size of the reflex angles below. ○

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R Find the size of the angles below by looking at the size of the reflex angle. d e

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180° angles

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180° degree angles are in fact straight lines.

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360° angles

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What does a 180° degree angle look like?

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This is the full way around the circle. In a circle all angles drawn will always add up to 360°.

R Using a protractor find the size of the angles in the circle below. a = ............. °

b = ............ °

c = ............. ° d = ............ ° e = ............. ° Now check if all the angles add up to 360°.

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ANGLES ON THE LINE Supplementary angles are two angles that add up to a total of 180°.

For example 120° + 60° = 180°.

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Complementary angles are two angles that add up to a total of 90°.

Note: All angles are contained within right angles (90°).

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R Find the size of the angles marked below.

R Fill in the size of the missing angles below.

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R Using the rules, calculate the size of the supplementary and complementary angles.

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WHICH ANGLE IS LARGER? R Look at the angles below. Without using a protractor, circle the letter of the largest angle.

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3. W = ...........................

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4. G = ............................

S = ...................................

5. R = .............................

Q = ................................

6. Z =.............................

U = ................................

7. E = .............................

V = .................................

8. N = ............................

P = .................................

9. T = .............................

K = .................................

R Using a protractor find the actual angle size for each of the angles above.

1. A = ............................

C = .................................

2. O = ............................

K = .................................

What word do the letters of the largest angles in each box spell out? ......................................

Write a definition for this word ...........................................................................................................

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NAMING ANGLES This angle has been labelled ∠QRS. The angle can also be known as angle R or ∠R.

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R Label the angles in each diagram below.

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Angles in Triangles Triangles have three angles.

This triangle has been labelled ∆MNO. The angles can be known as ∠M, ∠N, ∠O. Note: The triangle is known as MNO rather than ONM, in keeping with alphabetical order.

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N R Label the triangles below and then write the name of the largest angle underneath.

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Largest angle is ∠C

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Angles in Diagrams R In the diagram below all the points have been labelled.

F G In a diagram such as this, angles are not referred to by just one letter such as ∠J. Instead this angle is known as ∠KJM. The letter at the M point of intersection (vertex) goes in the middle.

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K H J R Write down the names of all the angles in this diagram. ....................................................

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MEASURING ANGLES 2 R 1. Measure angle ∠ABC using a protractor.

∠ABC = ___________

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R 2. Look at the angles in the diagrams below.

R Find the size of these angles.

∠GHI = .......................

∠LMN = .....................

∠QRS =.......................

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∠QSR = ......................

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R 3. Measure the angles listed under each triangle.

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∠JFB = ........................................

∠ CAR = .....................................

∠XYZ = ........................................

∠FBJ = ........................................

∠ RCA = .....................................

∠YXZ = ........................................

∠FJB = ........................................

ARC = .....................................

YZX = ........................................

R 4. Measure the angles in the diagrams below.

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∠TDP = .......................

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∠WDV =.....................

∠GEK = ......................

∠QOG = ....................

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ANGLES IN A TRIANGLE R 1. Measure each of the angles in these triangles. Add them together and write the total in the space below.

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a = ........................

a = .......................

b

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+ c = __________

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+ c = __________

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What did you find out? ..............................................................................................................................................

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R 2. Calculate the angle size of the triangles below.

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What do the angles of each of the triangles add up to? .................................................

Rule: The sum of the angles of any triangle is ...................................... °

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R 3. Using the rule, find the size of the angle x below without using your protractor.

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R 4. Solve the following problems by using the clues.

This triangle is an equilateral triangle and all angles are the same size. ∠WJA is a right angle ∠JAW = 47°

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∠ CRZ = .................................

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∠ AWJ = .................................

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ANGLE FACTS R Cut out four paper triangles on a spare piece of paper, based on these examples: Right Angle

Isosceles

Scalene

Equilateral

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R Label the angles of each triangle.

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R Carefully tear the corners off the right angle triangle and arrange them in a line like so:

R Repeat this process with the other three triangles and paste their angles below.

Isosceles

Scalene

Equilateral

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What did you find? .....................................................................................................................................................

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Try this activity with two different sized quadrilaterals.

R Paste the angles in the space below.

What did you find? .....................................................................................................................................................

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SCALENE TRIANGLES A scalene triangle is a triangle that has three sides of different lengths. It also has no equal angles.

R Measure the angles in these three triangles.

Write the angle sizes in the correct place in each triangle.

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R Draw three scalene triangles using a ruler and then measure each of the angles using a protractor.

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• R Circle the scalene triangles below. b.

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R List some objects in your classroom or playground that include a scalene triangle.

For example, the slide in the playground. ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... .........................................................................................................

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ISOSCELES TRIANGLES r o e t s Bo r e p ok u S

R 1. Write a rule about what you found.

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An isosceles triangle has two equal sides. Measure the angles in each of the isosceles triangles below.

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R 2. Without using a protractor, find the missing angles in the these isosceles triangles.

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R 3. Calculate the missing angles in the symmetrical house below without using a protractor. Use the other rules you know of to work out the angles.

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m = ................................. a = ................................. r = ................................. t = ................................. i = ................................. e = .................................

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EQUILATERAL TRIANGLES Equilateral triangles have three sides of equal length and angles of the same size.

Without measuring the equilaterals below, what size must each of the angles be? .....................................

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Equilateral triangles are always the same shape, just different sizes.

R Measure the angles in the triangles below to see if they are equilaterals and tick the ones that are.

You will have to be spot on with the protractor as some of them are pretty close! a.

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R Construct an equilateral triangle.

Draw a line in the space below.

Using a compass place the point on the left end of the line and the pencil exactly on the right end of the line.

Carefully mark a small line by swinging the compass around.

Make sure you do not change the width at which the compass is set.

Now repeat this action by placing the point of the compass on the other end of the line.

Join the ends of the line to the intersection of the two arcs (curves).

Measure the angle sizes using a protractor.

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GET THE RIGHT ANGLE

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A right triangle is a triangle that has one angle that measures exactly 90°.

It is often marked with a small square in the right angle.

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R Mark an X in the right triangles below. You may like to use a protractor to double check.

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The side of the right triangle that is directly opposite the right angle is known as the hypotenuse.

R Label the hypotenuse on each right triangle above and also the triangles below.

© ReadyEdPubl i cat i ons Pythagoras The famous Greek Pythagoras, 13s =e 5 s + 12o • f omathematician, rr evi e w pur po nl y• made an important discovery about right triangles.

169 = 25 + 144 2

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He found that, in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

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R Find the length of the hypotenuse in each of these triangles using this formula.

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R Challenge: Measure the length of sides a and b of this right triangle. Using a calculator, check to see what the hypotenuse should be using Pythagoras’ discovery.

Use your ruler to check if this is correct.

Try to be exact. Show the figures in this space. c2 = a2 + b2 Round your answer to two decimal places.

c.

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ANGLING AROUND So far we know that: All angles in a triangle add up to 180°.

The sum of any angles that make up a straight line is 180°.

All angles in a quadrilateral add up to 360°.

R Use these rules to find the missing angle x in the diagrams below.

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= .......................

x = .......................

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x = .......................

x = .......................

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∠MNK = 58° ∠ONL = 37°

x = .....................

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= .....................

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x = .....................

x = .....................

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INTERSECTING LINES R Measure the angles in the diagram below.

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∠ ACD =...........................................

∠BCE = ...........................................

∠DCE = ...........................................

∠ ACB = ..........................................

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R Complete this sentence to make up a new rule.

R Use this rule to find the size of angle x below.

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When two straight lines intersect ....................................................................................................... .

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x= 1. ...................... 2 ........................ 3........................ 4. ....................... 5. ....................... 6. .......................

Challenge: Find the values of all the other angles in each diagram above and write them in.

Remember the rule: Angles along a straight line add up to 180°.

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PARALLEL LINES In the diagrams below measure the size of angles x and y. Are they congruent?

x

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R Find the size of these angles by using all the rules that you know.

Remember: Angles made by intersecting lines are congruent.

You do not need a protractor for this activity.

110°

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Try out some more examples of your own on the back of this page.

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x

{ = ............°

= ............°

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x = ............°

R Measure the angles below.

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c

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d = ............°

e

= ............°

What did you find? .....................................................................................................................................................

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DEGREES IN A CIRCLE 1. R Using a protractor, measure the angles in each of the circles below.

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w = ..............°

d = ..............°

x = ..............°

e = ..............°

y = ..............°

f = ..............°

z = ..............°

g = ..............° h = ..............°

l = ..............°

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R Add the angle sizes for each circle.

m = ..............°

n = ..............° k = ..............°

j = ..............°

Total .............

Total ............. © R e a d y E d P u b l i c a t i o n s ...................° ...................° ...................° •f orr evi ew pur posesonl y• R Write a rule about what you found.

Total .............

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2. R Without using a protractor, find the missing angles below using the rules you know. Then check that all the angles add up to 360°.

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∠f = .................

∠g = .................

∠v = .................

∠j = .................

∠c = .................

∠r = .................

Remember: Angles that exist along a line are supplementary angles and always add up to 180°.

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CONSTRUCTING ANGLES 1 Angles can be constructed by using a protractor.

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1. R Construct the following angles using this method.

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To make an angle of 53°, place your protractor on the line PR below so that the middle point is at the end of the line (P) and the 0° mark points to R. Carefully find 53° on your protractor using the scale that starts at “0” and mark the point at the end. Rule a line to this point.

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23°

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These angles are all...................................................................... because they are less than ............................ °

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135°

178°

95°

These angles are all...................................................................... because they are more than......................... ° 2. R From the point in the lines below draw a line to construct an angle of the value underneath the line. Write the value of the other angle you have made.

These angles are known as ............................................. angles because they add up to ............................... °

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CONSTRUCTING ANGLES 2 It is possible to construct some angles without the use of a protractor. By carefully using a compass we can make exact angles using simple rules. The easiest angle to construct is probably a 60° angle.

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R On the line JK below, place the compass point at the J and stretch the pencil end to K. Without changing the width, draw an arc. Repeat this step using K as the compass point. Where these lines intersect draw a line. This angle should now measure exactly 60°.

To make a 30° angle we can bisect this 60° angle, again using only a compass. To bisect an angle or a line means to divide it into two equal sections.

Note: Make sure you do not change the width of the compass setting in your working.

arc

3. Where these arcs intersect draw a line to the vertex.

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1. Place the compass point on the vertex and draw an arc over the two rays.

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4. Measure the two new angles to check.

2. Where the arc intersects the rays a and b, place the compass point to draw two new arcs.

R Bisect this right angle to make a 45° angle using the method above.

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R Measure the angle below. Bisect this angle using your compass.

What are the sizes of the new angles? original angle = ............° new angles = ............°

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ANGLES AND DIRECTIONS 1 r o e t s Bo r e p ok u S

Find the treasure!

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R Look at the path in the grid. Follow the path and read the directions as you go.

The path starts at X and heads north for two squares.

A turn was then made to the left at a 90° angle and then after moving one square, another turn was made at a 90° angle to the right also moving one square. The next turn was a 45° turn to the left and moving along two squares.

A 45° turn to the left was then made and continued for only 1 square. A 90° turn to the left was made and the path moved 5 squares. Another 90° turn to the left was made and continued 2 squares to the end of the path.

R Now that you’re an expert in following directions find where the treasure is buried below.

Start at the X and move right for 3 cm and then turn 90° to the left. Continue for 4 cm and then make a 90° turn to the right and continue for 5 cm. Turn 45° to the right and travel 1 cm. Now turn 45° to the right and move 1 cm. Make a 90° turn to the right and move 8 cm. Again turn 90° to the right and move 3 cm. Turn left at a 90° angle and move 1 cm. Mark another X where the treasure is buried.

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Describe in words the path you would take from your classroom to the library giving the directions and approximate angles. For example, head south for 30 metres and make a right turn at the office.

Walk for 20 metres and then make a 45° turn to the left, etc. ......................................................................................................................................................................................... ......................................................................................................................................................................................... ......................................................................................................................................................................................... ......................................................................................................................................................................................... .........................................................................................................................................................................................

R Check your description with that of one of your friends and then draw a map on a scrap piece of paper based on your description.

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ANGLES AND DIRECTIONS 2 The compass below shows the angles of each direction.

R Label the following points on the compass.

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North(N), South(S), East(E), West(W), Northwest (NW), Southwest (SW), Northeast (NE), Southeast (SE).

© ReadyEdPubl i cat i ons 1. NW, N • NW,f W, N, E,r NE, N, i N,e SE, NE,p S, S, SE, E, S, W,s SW, S, SW. o r e v w u r p o e s onl y• Start at the X. R Follow the paths given using the grids below. Only move one square at a time.

2.

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2.

N, E, N, W, N, E, E, N, W, N, NE, NW, N W , NE, SE, NE, SE, NE, S, S, W, W, S, E, E, S, SW, NW, S, SE, S, W, S, E, S, W, W, W, W. R On a piece of 1 cm grid paper make a pattern and then give instructions for a friend to draw the pattern.

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SNOOKER ANGLES r o e t s Bo r e p ok u S

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Below is a diagram of a snooker table with only four pockets. The way in which it works is very simple. If you hit the ball it then bounces off the side cushion at a 90° angle until it finally lands in a pocket. In the diagram below, the track of the ball is mapped out for you.

Pocket

R Using this method, work out which pocket each of the following balls will go into if they are hit in the direction of the arrow. Draw a circle around the pocket.

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

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b.

d.

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ANGLE CHECK POINT R How many angles can you find? Use all the rules you have learnt so far to find the size of the missing angles in each diagram below. You should only need to use a protractor to check if you are correct. c. d.

x

b.

e.

z

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g.

x

x

i.

j.

x

x

k.

x

h.

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a.

z

l.

x

© ReadyEdPubl i cat i ons x x •f orr evi ew pur pos esonl y• x

n.

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x

{

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z x

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R Complete the following sentences.

1.

Supplementary angles add up to ........................ °

2.

Angles in an equilateral triangle are ........................................................................ .

3.

Opposite angles formed by two intersecting lines are ............................................................. .

4.

Angles in a circle add up to ....................... ° .

5.

Angles in a quadrilateral add up to ......................... ° .

6.

How many degrees are there in a straight line? ............................ °

7.

A right triangle is a triangle that has........................................................................................................... .

8.

A reflex angle is................................................................................................................................................ .

9.

Complementary angles add up to ...................... °

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ANGLES OF THE GAME The captain of the cricket team has asked you to draw a plan for the big match. He has specified that he would like each of the players to stand in a certain position when the other team is batting.

R Draw a plan of where the players should be standing by measuring the angles and marking them on the dotted line. You may like to draw some lines to help you mark each player off. Player 1 and player 2 have been added as examples.

r o e t s Bo r e p ok u S Player 2 - 55° to the right Player 4 - 85° to the right Player 6 - 30° to the right

In front of the batsman (x) Player 7 - 35° to the left Player 9 - 45° to the left Player 11 - Bowler

Player 8 - 55° to the right Player 10 - Wicket keeper

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Behind the batsman (x) Player 1 - 35° to the left Player 3 - 45° to the left Player 5 - 80° to the left

Hint: For some angles you may need to mark a dot at the edge of the protractor. You can then draw a line to show where the player would be.

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measure from here

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2 a © ReadyEdPubl i c t i ons X 1 X •f orr evi ew pur posesonl y•

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BASEBALL HITS Look at the baseball pitch below. The dots in the diamond indicate where the ball has landed. You may need to draw in some guide lines to help you measure.

r o e t s Bo r e p ok u S

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X batter

6

8°

7

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5

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3

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2

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R Find the angle of the shot for each of the hits above. The first one has been done for you.

8 1 = ..................................

0

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

3 = ..................................

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

5 = ..................................

6 = ..................................

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

8 = ...................................

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AN ANGLE ON TIME We see changing angles on a clock face every minute.

What sized angle is shown for the times below?

Hint: Use this clock face and pencil lines to help you.

5.30 - 15°

11.00 ...............

7.00 .................

8.30 .................

e.g.

Noon ..............

3.00 .................

r o e t s Bo r e p ok u S 9.00 .................

3.45 .................

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R Draw the times in the clock faces below using your ruler.

What angles are they showing?

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• 7.16 ................°

11.17 ................°

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Clock Quiz

1. How many degrees does an hour hand move between 11.00 and 12.00? ......................

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2. How many degrees are formed at exactly 6.00? ......................

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3. After 3.00, when is the very next time that the hands form a perfect right angle? ......................

4. What is the reflex angle size when a clock’s hands say 4.30? ......................

5. What kind of angle is formed by the clock hands at these times? Measure from the hour hand ......................................................................................................... e.g. 11.45 - reflex a. 3.45 ........................................

b. 11.30 .....................................

c. 9.33 ........................................

d. 7.45 ........................................

e. 12.45 ......................................

f. 2.17 ........................................

g. 1.55 ........................................

h. 5.58 .......................................

i. 12.37 ......................................

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ANGLES IN THE REAL WORLD (Hint : Draw diagrams where necessary.)

1. Mark is facing south. If he turns 90° to the right, what direction will he be facing?

2. Irene owns a triangular block of land. She knows that one corner is a 45° angle and the other corner is a right angle. What is the angle size of the third corner?

r o e t s Bo r e p ok u S

3. Katie was heading toward the markets in an eastern direction. She made a 90° turn to her right after 100 metres and then another 90° turn to her right about 200 metres further. In what direction was she now heading?

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4. Lesley noticed that the hands on the clock made a perfect 90° angle and that one of the hands was on the three. What time must it have been?

5. Denis walked 300 metres in a northern direction and made a 90° turn to the right. He then walked a further 250 metres and then made a turn south. How many degrees did he turn and in which direction.

6. Sarah was putting up a sail on her yacht. She noticed the sail was an isosceles triangle and that the smallest corner was a 30° angle. What must the sizes of the two other angles be?

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• What angle will the slide be from the ground

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based on these measurements? Use the box below, a ruler and a pencil to help you draw the model of the slide and then use a protractor to find the size of the angle.

1 box = 1 metre.

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7. The council is planning to build a new slide in the park. The ladder, which stands at 3 metres high when at right angles to the ground, will have a long slide attached to the top. The end of the slide is 7 metres from the bottom of the ladder.

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Angles for dinner?

Emily made a pizza for the family. She knew Carl would want the biggest piece and the size of the angle of his slice was 80°.

There were four other people that needed a piece.

If she made them exactly the same size what angle would each slice be at the point?

For dessert, a fresh apple pie was to be cut into 5 equal pieces.

What size angle should each piece be at the point if they are cut to exactly the same size? .................................

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PUZZLES WITH ANGLES R Use all the rules about angles and lines to help you solve the puzzles below.

Note: You do not need a protractor to solve the puzzles.

Lines on this page that look parallel are parallel and angles which look 90° are 90°.

r o e t s Bo r e p ok u S

Hint: Remember, isosceles triangles have two equal angles.

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R Find the size of every angle in the gates below. Only one angle is given.

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Ready-Ed Publications

PARTS OF A CIRCLE 1 In the circle below we have: a chord - CG a radius - AE a diameter - BF radii - DE and BE Note: Radii is the plural of radius.

r o e t s Bo r e p ok u S

In the circle below draw:

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A chord - YZ A radius - XW A diameter - RS Other radii - QW and TW

Measure the lengths of the radii in this circle and write them below.

© ReadyEdPubl i cat i ons XW = ............................

•f orr evi ew pur p osesonl y• QW = ............................

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Measure the diameter RS ................................

What do you notice? .......................................

In the circles below, name the radii and find their lengths.

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Z

MG - ..........................................

JB - .............................................

ZX - ...........................................

MI - ............................................

JF - .............................................

ZY - ...........................................

MR - ...........................................

JC - ............................................

ZW - ..........................................

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PARTS OF A CIRCLE 2 R Using a compass, ruler and a pencil draw circles with the following radii around the points below.

r o e t s Bo . . r e p ok u S

radius = 2 cm

radius = 3.5 cm

Arcs

An arc is a section of a curve with two end points.

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.

radius = 1.5 cm

© ReadyEdPubl i cat i ons C •f orr evi ew pur posesonl y•

R Look at the chord CD. The chord divides the circle into two arcs, the major arc which is longer and the minor arc which is shorter.

D

Arc CSD is the __________________ arc

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R Label the minor and major arcs of the circles below.

The first one has been done for you.

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and Arc CTD is the __________________ arc.

major arc = JEN

major arc = ..................................

minor arc = JAN

minor arc = ..................................

o c . che e r o t r s super major arc = ............................. minor arc = ............................

major arc = ............................. minor arc = ............................ major arc = ............................. minor arc = ............................

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TRIANGLES IN CIRCLES 1 R Join a line to the points to make the following triangles in the circles:

2. Triangle WTH

3. Triangle SJP

Is the triangle in the minor or major arc of the circle?

1. ......................................

2. ......................................

3.......................................

R Using a protractor find the size of each angle in the triangles.

∠ G= ..............................

∠ W =.............................

∠ S= ...............................

4. Triangle FMK

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1. Triangle GQR

r o e t s Bo r e p ok u S

4. .....................................

∠F = .............................

∠T = ............................. ∠ J= ............................... ∠M = ........................... © Re adyEdPubl i cat i ons ∠R = ............................. ∠H = ............................ ∠ P= .............................. ∠K = ............................

•f orr evi ew pur posesonl y• ∠Q = ............................

What do the angles of each triangle total? ..................................................

∆XYZ

∠B = .................. °

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∆ QRS

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R In the circles below draw and label the stated triangles and measure the angles. ∆ ABC ∠A = .................. °

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C = .................. °

∠X = ................. ° Y = ................. °

∠Z = .................. ° ∠Q = ................. ° ∠R = .................. ° ∠S = ................... ° Ready-Ed Publications

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TRIANGLES IN CIRCLES 2 R In the triangles below, name the arc in which the triangle has been drawn (major or minor) and decide whether the angle x is an acute, obtuse or right angle.

1. arc - .................................... angle - ...............................

r o e t s Bo x r e p ok u S x 2. arc - .................................

x

angle - .............................

3. arc - .................................. angle - .............................

4. arc - .................................. angle - .............................

5. arc - ..................................

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x

x

angle -............................. © Read y EdPubl i cat i ons •f orr evi ew pur posesonl y• 6. arc -.................................. angle - .............................

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angle - .............................

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7. arc - ..................................

x

x

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What did you find?

Angles formed in the minor arc using the end points of a chord are: ............................................................ .........................................................................................................................................................................................

Angles formed in the major arc using the end points of a chord are: ............................................................ .........................................................................................................................................................................................

Angles formed in a semicircle using the end points of the diameter are: ...................................................... .........................................................................................................................................................................................

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ANGLES IN CIRCLES 1 FI is a chord in the circle H. Measure angles FGI and FHI using a protractor.

∠FGI = ................................... °

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∠FHI = ................................... °

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What did you find?

In the circle K draw a chord LM. Draw a triangle in the major arc and label it LNM. Construct triangle KLM. Measure these angles:

∠LKM = ..................... ° © Rea dyEdPubl i cat i ons •f orr evi ew pur posesonl y•

∠LNM = ..................... °

R Make up a rule to describe what you found in both of the circles above.

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

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R In each circle below construct a triangle using the chord and the point in the major arc of the circle.

C

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T

A

B

A

B

A

B

R Measure the angle on the perimeter in each circle. What did you find? ......................................................................................................................................................................................... .........................................................................................................................................................................................

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ANGLES IN CIRCLES 2 R In the circle D join the points to make triangle CDE. R Make the triangle CFE in the minor arc.

R Measure the angles below using a protractor:

∠CDE = ________° ∠CFE = ________°

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

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What did you find? .....................................................................................

R In each circle below draw a chord and then construct two triangles - one using the chord and any point in the minor arc and one using the chord and the centre point. Label your diagrams.

© ReadyEdPubl i cat i ons •twof o rr v i e uyour r p osesonl y• R Measure the angles ine each of thew circlesp using protractor.

What kind of angle is the angle at the perimeter in each circle?

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R Make up a rule to describe what you found.

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

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R Without using a protractor, find the size of

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x in each circle below.

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SHAPES IN CIRCLES 1 Hexagons R Place the point of your compass in the centre.

Set your compass so that the pencil touches the perimeter and then move the compass point to any point on the perimeter.

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Mark off points on the perimeter until you have six marks evenly spaced around the circle.

Rule a line from mark to mark to form the hexagon.

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Measure the angles of the hexagon...............................

Are the angles congruent? ...........................

Squares

R Using your compass, place the point on one end of the diameter and make an arc at the top of the circle and the bottom of the circle. Repeat this by placing the point at the other side.

© ReadyEdPubl i cat i ons Joins the two where arcs •f orr evi ew pur po e spoints on l ythe• intersect to make a second diameter of the circle.

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Now join the ends of the diameter in the circle to form the square.

R Measure each angle in the square. If you were exact with the construction, each angle should be ..................... °

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R Using a compass, construct some symmetrical shapes or patterns in the circles below.

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SHAPES IN CIRCLES 2 R Measure each of the angles in the circle below using a protractor.

What size is each angle? ............................. ° Multiply this number by the number of angles (5).

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Join up the points where the radii meet the edge of the circle. What shape have you formed?

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.............................. ° × 5 = .......................... °

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

Knowing that all angles in circles add up to 360°, construct the following regular shapes in the circles below by first working out how many degrees there are in each angle of the shape. You will need to be exact with the angles so that the shapes are regular.

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Hexagon

Nonagon (9 sides) Page 40

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N.B. you do not need a compass for this activity.

o c . che e r o t r s super Equilateral Triangle

Decagon (10 sides)

Octagon

Dodecagon (12 sides) Ready-Ed Publications

ANSWERS Looking at Different Angles (page 5) Right angles - 4, 6, 9; obtuse angles - 3, 5, 7; acute angles - 1, 2, 8.

Measuring Angles 1 (page 6) 3. 20° a - 63°, acute b - 30°, obtuse f - 165°, obtuse g - 28°, acute

c - 26°, acute h - 90°, right d = 60° d = 70°

e = 70° e = 52°

f = 55°

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Angles on the Line (page 8) Supplementary angles - 90°, 30°, 55°.

Complementary angles - 33°, 45°, 6°, 27°.

102°, 58°, 45°.

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e - 60°, acute

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Reflex Angles (page 7) 1. a = 335° b = 240° c = 340° 360° angles a = 65° b = 63° c = 110°

x = 80°

d - 90°, right i - 120°, obtuse.

Which Angle is Larger? (page 9) N.B. Please note angles may vary slightly due to the printing process.

These answers should be used as a guide.

1) A = 47°, C = 56° 2) O = 109°, K = 94° 3) W = 16°, N = 25°

4) G = 133°, S = 120°

5) R = 110°, Q = 70°

8) N = 71°, P = 45°

6) Z = 60°, U = 139°

7) E = 80°, V = 60°

Congruent. © R eadyEdPubl i cat i ons Naming Angles (page 10) Answers will vary. ∠FHI, ∠FHG, ∠GHJ, ∠HGJ, ∠GJM, ∠MJK. •f orr evi ew pur posesonl y• Measuring Angles 2 (page 11)

9) T = 90°, K = 87°.

3.

∠JFB = 60° ∠FBJ = 60° ∠CAR = 105° ∠RCA = 45° ∠XYZ = 65° ∠YXZ = 43°

4.

∠TDP = 80°

2. ∠GHI = 135°

∠LMN = 65°

∠FJB = 60° ∠ARC = 30° ∠YZX = 72°

∠WDV = 180° ∠GEK = 20°

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Angles in a Triangle (page 12) 1. a = 25° b = 130° c = 25°.

2.

They should all add up to 180° 3. x = 53° x= 39° x = 68° 4. ∠CRZ = 60° ∠AWJ = 43°.

∠QRS = 62°

a = 60°

∠QOG = 45°. b = 45°

∠QSR = 64°.

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∠ABC = 42°

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1.

c = 75°.

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Angle Facts (page 13) Checks diagrams on page.

Scalene Triangles (page 14) Check diagrams of triangles. Scalene - b, f, g.

Isosceles Triangles (page 15) 1.

An isosceles triangle always has two congruent angles at the base of the two congruent sides.

2.

a) g - 81°, m - 18 b) g - 68°, m - 44 c) p - 75°, q - 30°

3.

s - 130°, m - 25°, a - 58°, r - 58°, t - 32°, i - 32°, e - 58°

Equilateral Triangles (page 16) 60°. Equilateral triangles - a, e.

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Get the Right Angle (page 17) Check diagrams on sheet. Pythagoras - a. 10 cm, b - 5 cm, c - a= 2 cm, b = 6 cm, c ≈ 6.32.

Angling Around (page 18) 1)

x=123°

2)

x=69°

3)

x=133°

4)

x=27°

5)

x=85°

6)

x=58°

7)

x=40°

8)

x=30°

Intersecting Lines (page 19) ∠ACD = 30° ∠BCE = 30° ∠DCE = 150° ∠ACB = 150°. The opposite angles are equal in size.

1.

x=125°

2.

x=37°

3.

x=117°

4.

x=45°

5.

x=94°

6.

x=90°

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Parallel Lines (page 20) x and y are congruent. w = 105° z = 105° x = 110° y = 110° a = 45° b = 135° c = 150° y = 30° Each pair of angles within the parallel lines adds up to 180°.

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2. f = 125° j = 65° A circle has 360°.

g - 180°

z - 90° m - 105°

d - 45° n - 30°

e - 45° k - 45°

c - 55°

v = 46°

r = 115°

f - 90° j - 135°

Constructing Angles 1 (page 22) 1. Check student’s diagrams.2. Supplementary, 180°.

The angles should measure 141°, 23°, 64°.

Constructing Angles 2 (page 23) Check diagrams.

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Degrees in a Circle (page 21) 1. w - 90° x - 90° y - 90° g - 135° h - 45° l - 45°

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Angles and Directions 2 (page 25)

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© ReadyEdPubl i cat i ons Angles and Directions 1 (page 24) •f orr evi ew pur posesonl y•

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Snooker Angles (page 26) a) lower left-hand pocket, b) top right-hand pocket, c) top right-hand pocket, d) lower left-hand pocket, e) top left-hand pocket, f) top left-hand pocket.

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Ready-Ed Publications

Angle Check Point (page 27) a) x=50° b) x=80°, z=80° c) x=130°, z=50° d) x=60° e) x=90° f) x=120° h) x=335° i) x=309° j) x=55° k) x=70°, z=40° g) x=85°, z=95°, {=95° l) x=25° m) x=120°, z=60°, {=60° n) x=87°, z=87°, {=93° o) x=145°, z=35° 1) 180° 2) congruent 7) an angle of exactly 90°

3) congruent 4) 360° 5) 360° 8) an angle between 180 and 360°

6) 180° 9) 90°

Angles of the Game (page 28) Check diagram.

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Baseball Hits (page 29) N.B. - Answers are approximate - 1) 8° 97° 8) 46°

2) 31°

3) 16°

4) 43°

5) 61°

6) 66°

7)

Clock Quiz

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An Angle on Time (page 30) Answers may vary, check diagrams.

1) 30° 2) 180° 3) 3.33 4) 315° 5) a) obtuse b) obtuse c) reflex d) acute e) reflex f) acute g) reflex h) obtuse i) reflex.

Angles in the Real World (page 31) 1) West 2) 45° 3) West 4) 3.00 5) He turned right 90° 5) Both angles are 75° 7) Check diagram - angle should be 25°. The slide angle will be 23°.

Angles for Dinner? - pizza -70°, pie -72°.

© ReadyEdPubl i cat i ons Parts of a Circle 1 (page 33) Check diagram.

All will be 4i cm.

Diameter 8r cm.

•f oradii rr e v ew p=u posesonl y• Puzzles With Angles (page 32) Check diagrams.

Circle M - all radii are 1.5 cm, Circle J - all radii are 2.3 cm, Circle Z - all radii are 2 cm.

Triangles in Circles 1 (page 35) 1. major, 2. major, 3. minor, 4. major.

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Parts of a Circle 2 (page 34) Check circle diagrams. Arcs - Arc CSD is the minor arc and arc CTD is the major arc. Answers will vary depending on labelling.

Answers will vary depending on where students connect to points. The answers below are approximate.

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∠G= 40° ∠W= 50° ∠S= 50° ∠F = 70°

∠Q = 80° ∠T = 80° ∠J= 100° ∠M = 50° ∠R = 60° ∠H = 50° ∠P= 30° ∠K = 60° They all total 180° .

Check diagrams and angles.

Triangles in Circles 2 (page 36) 1. major arc, acute; 2. minor arc, obtuse; 3. neither, right angle; 4. minor, obtuse; 5. neither, right angle, 6. major, acute; 7. major, acute, 8. neither, right angle.

Students should find that angles formed in the minor arc are obtuse, angles formed in the major arc are acute and that all angles formed using the end points of the diameter are right angles.

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Angles in Circles 1 (page 37) ∠FGI = 60° ∠FHI = 120°.

Answers will vary according to diagrams. The outer angle is half the size of the central angle.

Angles in Circles 2 (page 38) Answers may vary according to drawings. The angles CDE and CFE should be congruent.

x = 70°, x = 60°, x = 70°, x = 45°.

Shapes in Circles 1 (page 39) Hexagons - the angles should be congruent; Squares - each angle should be 90°.

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Shapes in Circles 2 (page 40) 72°, 72 X 5 = 360, pentagon.

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