australian curriculum mathematics year 7 · review 1.1 lines, rays and segments a line continues in...

AUSTRALIAN CURRICULUM

MATHEMATICS YEAR 7

Angles

MATHEMATICS YEAR 7

Angles

Student’s name: ________________________________

Teacher’s name: ________________________________

First published 2012

ISBN 9780730744474

SCIS 1564097

© Department of Education WA 2012 (Revised 2020)

Requests and enquiries concerning copyright should be addressed to:

Manager Intellectual Property and Copyright Department of Education 151 Royal Street EAST PERTH WA 6004

Email: [email protected]

This resource contains extracts from The Australian Curriculum Version 3.0 © Australian Curriculum, Assessment and Reporting Authority 2012. ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that:

the content descriptions are solely for a particular year and subject

all the content descriptions for that year and subject have been used

the author’s material aligns with the Australian Curriculum content descriptions for the relevant year andsubject.

You can find the unaltered and most up to date version of this material at www.australiancurriculum.edu.au. This material is reproduced with the permission of ACARA.

Graphics used in this resource are sourced from http://openclipart.org under the creative commons license http://creativecommons.org/publicdomain/zero/1.0

This product will be registered through the National Copyright Unit for use in all Australian schools without remuneration.

https://creativecommons.org/licenses/by-nc/4.0/

creativecommons.org/licenses/by-nc-sa/3.0/au/

Year 7 Mathematics Angles

© Department of Education WA 2012 – MATHSAC027 Page 1

Contents

Signposts ....................................................................................................................................2

Introduction...............................................................................................................................3

Curriculum details....................................................................................................................4

1. Angle naming conventions .................................................................................................7

2. Classifying angles..............................................................................................................15

3. Adjacent and vertically opposite .....................................................................................23

4. Complementary and supplementary...............................................................................29

5. Perpendicular and parallel lines......................................................................................37

6. Angles and parallel lines...................................................................................................45

7. Using angle relationships..................................................................................................51

8. Summary............................................................................................................................59

9. Review tasks ......................................................................................................................61

Solutions...................................................................................................................................69

Angles Year 7 Mathematics

Page 2 © Department of Education WA 2012 – MATHSAC027

Signposts Each symbol is a sign to help you.

Here is what each one means:

The recommended time you should take to complete this section.

An explanation of key terms, concepts or processes.

A written response. Write your answer or response in your journal.

Correct this task using the answers at the end of the resource.

Calculators may not be used here.

Make notes describing how you attempted to solve the problem. Keep these notes to refer to when completing the Self-evaluation task. Your teacher may wish you to forward these notes.



Introduction This resource should take you approximately two weeks to complete. It comprises seven learning sections, a summary section and a review task section.

The learning sections have the following headings:

Key wordsThese are the main words that you need to understand and use fluently to explain yourthinking.

Warm-upWarm-up tasks should take you no longer than 10 minutes to complete. These are skills from

previous work you are expected to recall from memory, or mental calculations that you are expected to perform quickly and accurately. If you have any difficulties in answering these questions, please discuss them with your teacher.

ReviewSome sections have reviews immediately after the warm-up. The skills in these reviews are

from previous work and are essential for that section. You will use these to develop new skills in mathematics. Please speak to your teacher immediately if you are having any trouble in completing these activities.

Focus problemFocus problems are designed to introduce new concepts. They provide examples of the types

of problems you will be able to solve by learning the new concepts in this resource. Do not spend too long on these but do check and read the solutions thoroughly.

Skills developmentThese help you consolidate new work and concepts. Most sections include skills developmentactivities which provide opportunities for you to become skilled at using new procedures, applyyour learning to solve problems and justify your ideas. Please mark your work after completingeach part.

Correcting your work

Please mark and correct your work as you go. Worked solutions are provided to show how you should set out your work. If you are having any difficulty in understanding them, or are getting the majority of the questions wrong, please speak to your teacher immediately.

Journal

Please keep an exercise book to record your notes and to summarise your learning. At the end of each section, write definitions for the key words that were introduced for that section.



Curriculum details Content Descriptions

This resource provides learning and teaching to deliver the Australian Curriculum: Mathematics for the following Year 7 Content Descriptions.

Identify corresponding, alternate and co-interior angles when two straight lines are crossed by a transversal (ACMMG163)

Investigate conditions for two lines to be parallel and solve simple numerical problems using reasoning (ACMMG164)

Content Descriptions 1 2 3 4 5 6 7 R

ACMMG163

ACMMG164

Indicates the content description is explicitly covered in that section of the resource.

Previous relevant Content Descriptions

The following Content Descriptions should be considered as prior learning for students using this resource.

At Year 6 level

Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles (ACMMG141)

Proficiency strand statements at Year 7 level

At this year level:

Understanding includes describing patterns in uses of indices with whole numbers, recognising equivalences between fractions, decimals, percentages and ratios, plotting points on the Cartesian plane, identifying angles formed by a transversal crossing a pair of lines, and connecting the laws and properties of numbers to algebraic terms and expressions

Fluency includes calculating accurately with integers, representing fractions and decimals in various ways, investigating best buys, finding measures of central tendency and calculating areas of shapes and volumes of prisms

Problem Solving includes formulating and solving authentic problems using numbers and measurements, working with transformations and identifying symmetry, calculating angles and interpreting sets of data collected through chance experiments

Reasoning includes applying the number laws to calculations, applying known geometric facts to draw conclusions about shapes, applying an understanding of ratio and interpreting data displays



General capabilities

General capabilities 1 2 3 4 5 6 7 R

Literacy

Numeracy

Information and communication technology (ICT) capability

Critical and creative thinking

Personal and social capability

Ethical behaviour

Intercultural understanding

Indicates general capabilities are explicitly covered in that section of the resource.

Cross-curriculum priorities

Cross-curriculum priorities 1 2 3 4 5 6 7 R

Aboriginal and Torres Strait Islander histories and cultures

Asia and Australia’s engagement with Asia

Sustainability

Indicates cross-curriculum priorities are explicitly covered in that section of the resource.

This resource contains extracts from The Australian Curriculum Version 3.0 © Australian Curriculum, Assessment and Reporting Authority 2012. ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that:

the content descriptions are solely for a particular year and subject

all the content descriptions for that year and subject have been used

the author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject.

You can find the unaltered and most up to date version of this material at www.australiancurriculum.edu.au. This material is reproduced with the permission of ACARA.



1. Angle naming conventionsWhen you complete this section you should be able to:

correctly identify and name angles.

Key words

line ray line segment angle vertex arc

Warm-up 1

1. List the factors of 15. ____________________

2. 9 + 9 = _________

3. What is the missing number?

a = __________

4. Circle the greater fraction. 2

3or

3

5

5. Find three-halves of 10. __________

6. 9.3 + 2.4 = ____________

7. 2 . 8

6

8. Write 0.875 as a fraction. ____________

9. Complete: 67, 74, 81, ________

10. Determine the probability the spinner will land on an even number.

Express your answer as a fraction.

_________________

12

345

6

7

8

-4 -3 -2 -1a



Review 1.1

Lines, rays and segments A line continues in both directions. To show this, lines should be drawn with arrows at both ends.

The notation for the line shown is AB

.

A ray starts at a point and then continues off in a certain direction.

The notation for the ray shown is PQ

.

A line segment or interval is part of a line. It starts at a point and ends at a point.

The notation for the line segment shown is EF .

Example Write the correct notation for the following.

(a)

(b)

Solution

(a) JK

(b) MN

1. Write the correct notations for each of the following.

(a) ___________

(b) ___________

(c) ___________

2. Draw ST

in the space provided.

A B

QP

FE

QR

KJ

NM

D C

B A

Note how the ray is named. A ray starts from the point and

moves away, so the convention is that it should be

named with the point first.



Review 1.2

The size of an angle is the measure of turn or rotation. For example, the following shows the angle between the two hands of a clock increasing from left to right.

Angles can be represented using diagrams like the one that follows.

Example Place the angles in order of size from smallest to largest.

I. II. III.

Solution The angles from smallest to largest are III, I and II.

The size of an angle is not determined by the size of its diagram. Just because the arms of

the angle or the arc marking the angle is large does not mean the angle is large.

Remember, an angle’s size is a measure of rotation.

arc (showing the rotation)

armvertex



1. For each of the following pairs of angles, tick the larger angle.

(a)

(b)

(c)

2. Using the terms vertex, arm and arc, label the parts of the angle shown.

3. The following represents angle A.

In the space below, draw an angle:

(a) smaller than angle A (b) larger than angle A.

A



Focus problem 1

Helen was excited at getting her new tablet’s keyboard extension. She explained to her friend, Mai, that her tablet could rotate 240 from the keyboard extension.

Mai disagreed and said that it would only rotate 120. To prove her point, Mai drew the following diagram to represent the situation, saying that KBT was the limit of the rotation.

Helen became confused as she didn’t know what KBT represented.

Draw an arc on the diagram above to show the angle that Mai is trying to show.

Check your work before continuing.

Mobile devices

According to some sources, by 2016 it is predicted there will be over 1.4 mobile devices, like tablets and smart phones, for every person in the world! Currently the world’s population is around 7 billion, so using that population figure, it represents at least 9.8 billion mobile devices.

H

T

KB



Skills development 1.1

Angle notation When two or more lines, line segments or rays intersect, angles are formed about the point of intersection. This point is known as the vertex.

There are a few different ways of labelling angles. For example, the first angle above could be labelled ABC , CBA or B . Note how the angle vertex is in the middle of the string of letters.

An arc can be used to indicate the angle. For example, MQN and NQO are shown below.

Example Write the correct notation for the following angle as indicated by the arc.

(a)

(b)

Solution (a) KLJ , JLK or L

(b) DBC or CBD

AB

C

FD

E

OM

N

P

Q

LJ

K

OM

N

P

Q OM

N

P

Q

C

AB

D

Note that B would not be an appropriate name for the angle in part (b), as it could represent one

of at least three angles.



1. Write the three ways of labelling the following angle.

___________ ____________ __________

2. Write the correct notations for each of the following angles indicated by the arc.

(a)

______________________

(b)

______________________

(c)

______________________

(d)

______________________

3. Use an arc to mark each of the given angles. Assume that they are the smaller size.

(a) ABD

(b) BEC

N O

M

I F

E

G

H

C

AB

D

N O

M

FD

E

CA

B

D

CA

B

D

E




Other angle notation Another way to identify an angle is to use a lower case letter or symbol. For example, the following shows the size of ACB identified using the lower case letter, x.

Example Name the angle identified by the lower case letter.

(a)

(b)

Solution (a) OQP , PQO or Q

(b) DBA or ABD

1. Name the angle identified by the lower case letter.

2. What lower case letter is at SNR ? (Assume that it is the smaller angle.)


BE

A

DC

x°

QP

O

g°

C

AB

D

y°

Remember B would not be an appropriate name for the angle in

part (b).

A

C

B

h°

D

S

Q

P

a°

R

T

b°c°

d° N

Actually, the letter x is a variable. That is, it

represents a number. This is why it is italicised.



2. Classifying anglesWhen you complete this section you should be able to:

measure and classify angles.

Key words

protractor angle vertex degree acute angle obtuse angle reflex angle right angle straight angle revolution

Warm-up 2

1. Circle the largest common factor of 3 and 6. 1, 2, 3, 6, 12

2. 14 – 7 = _________

3. The temperature was 6 degrees but it dropped 10 degrees.

What is the new temperature? _________

4. Insert <, > or = to make the following sentence true. 1 2

3 6

5. 1

215 = _________

6. Round 8.62 to a whole number. __________

7. 52.68 2 = _________

8. Write 37% as a decimal. _________

9. Complete: 2.6, 2.8, 3.0, ________

10. Determine the size of the missing angle.

? 97°

0

5

10

20

°C



Review 2

A protractor can be used to measure the amount of rotation.

To correctly measure angles using a protractor, follow these steps.

1. Place the centre point of the base of the protractor on the vertex of the angle.

2. Rotate the protractor so that its base line is on one of the arms of the angle. Make sure theprotractor is covering the angle. If it is not, then rotate the protractor until it does. Also makesure that the centre point is still on the vertex.

3. Read off the angle size on the protractor from where the angle’s second arm points to the scale.Make sure you read from the correct scale.

Example Measure the following angle.

Note that there are two scales on the protractor, an inside and an outside scale. The scale you use depends on how you

line your protractor up against your angle.

0 180

10

170

20

160

30150

4014050

13060

12070

11080

10090100

80

110

70

12060

13050

140

4015

03016

02017

010

1800

Centre point (of the base)

Outside scale Inside scaleBase line

Note how this angle has no arrows. The arrows

are often left off for convenience.



Solution Step 1 – Place the centre point of the base of the protractor on the vertex of the angle.

Step 2 – Rotate the protractor so that its base line is on top of one of the arms of the angle.

Step 3 – Read the size of the angle from the correct scale.

The size of the angle is 140.

0

180

10170

2016030

15040

14050

13060

120

70

110

80

100

90100

80

11070

120

6013

05014

04015

030

160

20

170

10

180

0

0 180

10

170

20

160

30150

4014050

13060

12070

11080

10090100

80

110

70

12060

13050

140

4015

03016

02017

010

1800

0

180

10170

2016030

15040

14050

13060

120

70

110

80

100

90100

80

11070

120

6013

05014

04015

030

160

20

170

10

180

0

The arm points to here.

The protractor's base line is on top of this arm so the

inside scale is used.

Note that you can place the base line of the protractor on either arm as long as it

covers the angle. You should get the same result.

Sometimes you might have to extend the angle’s arms so that you can position and

read off the protractor correctly.

Also, note how the degree sign is used.



1. Estimate the size of each of the angles shown below then measure the size correct to the nearestdegree.

(a)

Estimate: _______________

Measured size: ___________

(b)

Estimate: _______________


(c)

Estimate: _______________


(d)

Estimate: _______________


(e)

Estimate: _______________




Focus problem 2

Alex was having trouble determining which scale to use on his protractor. Jenni suggested classifying the angles first to determine if they are bigger or smaller than 90.

Without measuring, decide if each of the following angles are greater or less than 90. Write you answer below each one of them.

_______________________ _______________________


Kicking at goals

A recent addition to the AFL football coverage has been the graphic depicting the angle and probability of kicking for goal. The angle is measured from the perpendicular, so zero degrees is a shot taken directly from in front of the goal. An angle such as 80 degrees would be almost impossible and 90 degrees would be in line with the posts. Next time the footy is on, tell your folks you are watching it for maths!

A B

0 180

10

170

20

160

30150

4014050

13060

12070

11080

10090100

80

110

70

12060

13050

140

4015

03016

02017

010

1800



Skills development 2

Angles are measured in degrees and can be classified according to their size.

An acute angle is between 0 and 90.

A right angle is exactly 90.

An obtuse angle is between 90 and 180.

A straight angle is exactly 180.

A reflex angle is between 180 and 360.

A revolution is exactly 360.

Example Classify the following angles as either acute, obtuse, right, straight, reflex, or revolution.

(a)

(b)

(c)

Solution (a) It is an obtuse angle as it is between 90 and 180.

(b) It is an acute angle as it falls between 0 and 90.

(c) It is a right angle. Although it does not show its measurement, it is exactly 90. This is knownbecause of its special marking.

1. (a) An acute angle is between zero degrees and ______degrees.

(b) An obtuse angle is between 90o and ______o.

(c) A right angle is exactly ______o.

(d) A straight angle is exactly ______o.

(e) A revolution is an angle that is ______o.

(f) A reflex is an angle that is greater than 180o but less than ______o.

Note how the degree

sign is used.

124°

72°

Where is the angle?

Read the answer

below to find out.



2. Classify each of the following angles as either acute, obtuse, right, straight, reflex, orrevolution.

(a)

(b)

(c)

(d)

(e)

(f)

(g)


146°

56°

146°

180°

214°



3. Adjacent and vertically oppositeWhen you complete this section you should be able to:

identify adjacent and vertically opposite angles.

Key words

adjacent vertically opposite

Warm-up 3

1. Circle the composite numbers. 3, 5, 7, 9

2. 6 4 = _________


a = _________

4. Locate 3

4 on the number line.

5. Find two-thirds of 15. _________

6. Estimate the difference by first rounding to whole numbers.

38.3 – 21.1 ___________

7. 8.21 3 = _________

8. Write 1

6 as a percentage. _________

9. Complete:1 2 3

6 6 63 , 4 , 5 , _________

10. The truck is shown at (-3, 2).

If the truck moves 3 units right, where will it then be?

_________ x-5 -4 -3 -2 -1 1 2 3 4 5

y

-3-2-1

123

0 1

-5-8 1 4a



Focus problem 3

As part of the company’s requirements, Stan and Patricia were asked to survey the intersection of two roads, as shown below. The company specifically wanted to know the angles between the two roads.

Stan measured one of the angles between the two roads and recorded it on a diagram, as shown below. He then told Patricia that he would work the rest out when he got back to the office.

How can Stan determine the rest of the angles and what would they be?


Surveying

Surveyors accurately determine distances and angles between points. Today, they use computers, laser rangefinders and satellite-positioning systems to create real-time surveys. These surveys can immediately be used to make decisions about construction etc.

62°




Adjacent angles are next to one another. They have a common arm (line, line segment or ray) and a common vertex.

In the example below, ADB is adjacent to BDC .

Example Determine the acute angle which is adjacent to RUS .

Solution Angle SUT is the acute angle that is adjacent to RUS .

Other angles are adjacent but not acute. For example, RUT is adjacent to RUS but it is not an acute angle.

1. On the diagram, mark the obtuse angle that is adjacent to EHF .

2. Consider the diagram and complete the following.

(a) The angle marked with a is adjacent to ____ and ____.

(b) The angle marked with b is adjacent to ____ and ____.

(c) The angle marked with c is adjacent to ____ and ____.

(d) The angle marked with d is adjacent to ____ and ____.

(e) The angle marked with e is adjacent to ____ and ____.

a°

b°c°

e°

d°

TRU

S

E

HF G

D

AB

C

common armcommon vertex




Vertically opposite angles are formed when two lines intersect. In the example below, AED and BEC are vertically opposite.

AEB and CED are also vertically opposite.

Vertically opposite angles are congruent. That is, their sizes are equal.

Example Determine the angle vertically opposite BEC .

Solution Angle AED is vertically opposite to BEC .

1. Mark the angle that is vertically opposite PRT .

2. Using your own words, define ‘congruent’.

_______________________________________________________________________

A B

CD

E

A

B

C

DE

A

B

C

DE

P

Q

S

T R



3. Compare the sizes of the four angles marked below. You may like to measure their sizes with aprotractor.

(a) Circle the correct statement.

(i) a°

is greater than

is the same size as is less than

b°

(ii) x°

is greater than

is the same size as is less than

y°

(b) Will this always happen? Give your reasoning.

____________________________________________________________________

____________________________________________________________________

____________________________________________________________________

4. The following shows three lines intersecting at a point.

Compare the angles in the diagram above. Determine if the following statements are true.

(a) m° = n° (Yes/No)

(b) a° = b° (Yes/No

(c) g° = h° (Yes/No)

a°

b°

y°

x°

a°

b°m°

n°g°h°

Notice that an extra line creates more vertically opposite angles.



5. Complete the sentence.

Vertically opposite angles are e_________ in size. That is, they are congruent.

6. Determine the sizes of the angles marked with a letter in each of the following.

(a)

(i) x = __________

(ii) y = __________

(b)

(i) m = __________

(ii) n = __________

(c)

(i) a = __________

(ii) b = __________

(iii) c = __________


y°

x°

140°

40°

n°

m°65°

115°

b°

a°

c°

30°

60°



4. Complementary and supplementaryWhen you complete this section you should be able to:

identify complementary and supplementary angles.

Key words

complementary angles supplementary angles

Warm-up 4

1. Fill in the missing value. 15, 10, 6, ______ , 1

2. 36 4 = _________

3. The temperature was minus 6 degrees but it went up 10 degrees.

What is the new temperature? _________

4. Express the value of w as a fraction.

5. 3

416 = _________

6. 0.201 1000 = _________

7. 5 4.05

8. Write 331

3% as fraction. _________

9. Complete: 213, 208, 203, _________

10. A six-sided die is rolled.

Express, as a fraction, the probability that it lands on a composite number.

_________

10

w

0

5

10

20

°C



Focus problem 4

The efficiency of a solar panel is based upon the angle of the sun to its surface.

While studying this effect, Ella determined that the sun was at an angle of 50 to her solar panel, as shown in the diagram below. For her calculations, Ella wanted to find the adjacent angle to this.

Determine the size of the missing angle.


Solar technology

Western Australia is above the latitude of 36S. Its climate is such that the sun is visible most days of the year. However, the amount of energy generated by renewable energy sources, like solar, is small compared to countries like Germany, which sits above the 45N latitude line.

50°

What is the size of this angle?



Skill Development 4.1

Angles which add up to 90 are called complementary angles.

For example, the angles marked with m and n are complementary angles.

Example Determine the size of the missing angle.

Solution a = 90 − 30

= 60

The missing angle is 60.

1. Decide whether these pairs of angles are complementary.(Write complementary or not complementary under each pair.)

(a)

____________________

(b)

____________________

(c)

____________________

m°n°

a°

30°

We can use the fact the angles are complementary to find the size of the missing

angle.

We know that a + 30 = 90, so therefore 90 – 30 = a.

25°

65°

So, m + n = 90.

47°

43°64°

46°



(c) 24 and 56

____________________

(d)

____________________

(e)

____________________

2. Find the size of y in each of the following diagrams.

(a)

y = __________

(b)

x = __________

3. Complete these sentences.

(a) The complement of 78° is _________.

(b) The complement of 9° is _________.

(c) An angle of 23° is the complement of an angle of _________.

71°19°

30°70°

y°

56°

x°

29°



Skill Development 4.2

Angles which add up to 180 are called supplementary angles.

For example, the angles marked with r and s are supplementary angles.

Example Determine the size of the missing angle.

Solution A = 180 − 70

= 110

The missing angle is 110.

1. Decide whether these pairs of angles are supplementary.(Write supplementary or not supplementary under each pair.)

(a)

_______________________________

(b)

_______________________________

(c) 124 and 56

_______________________________

(d)

_______________________________

r° s°

We can use the fact the angles are supplementary to find the size of the missing

angle.

We know that a + 70 = 180, so therefore 180 – 70 = a.

So, r + s = 180.

47°133°

a° 70°

27° 126°

60°120°



2. The following two angles are supplementary because they add to __________________

3. Complete these sentences.

(a) The supplement of 30° is _________.

(b) The supplement of 125° is _________.

(c) An angle of 103° is the supplement of an angle of _________.

4. The following angles are supplementary.

If x = 85°, what is the size of angle y? _________.

118° 62°

If these two angles above were joined

together they would form a straight line.

y°x°

Oh, and a straight angle measures 180.



5. In this diagram, one of the angles is 90°.

(a) What part of the diagram lets you know that one of the angles is 90?

____________________________________________________________________

(b) What are the sizes of the other angles?

a° = __________

b° = __________

c° = __________


a°

b°c°



5. Perpendicular and parallel linesWhen you complete this section you should be able to:

identify perpendicular and parallel lines.

Key words

perpendicular lines parallel lines

Warm-up 5

1. Express 6 as factors of primes. ____________

2. 37 + 4 = _________


u = __________

4.6 8

10 10

5. Find three-quarters of 24. __________

6. 0.708 km = __________ m

7. 8 – 6 2 = _________

8. Write 1

64 as decimal. _________

9. Complete: 62.9, 58.6, 54.3, ________

10.

Which shape is at (-2, 0)?

__________ x-5 -4 -3 -2 -1 1 2 3 4 5

y

-3-2-1

123

-2-14 -6-10 u



Focus problem 5

While staying at a holiday resort, the Parkinson’s family were told that the local surf club serves up a meal every Sunday. Not knowing the area, Mr Parkinson asked for directions to the club.

He was given the following two directions and became a little confused.

From Gate 1:

Follow the road until you find one that runs perpendicular to it. Turn into this road and followit until the end.

From Gate 2:

Take either road to the right, as they run parallel to each other. Go to the end and turn left.

Surf club

Holiday resortGate 1

Gate 2



Help Mr Parkinson find the direction to the surf club by explaining the term perpendicular and parallel.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________





When two lines intersect at right angles (90°), the lines are perpendicular.

Example Which of these pairs of lines are perpendicular?

I. II. III.

Solution II and III show lines that are perpendicular.

1. Which of these pairs of lines are perpendicular? Circle the correct response.

(a)

Yes / No

(b)

Yes / No

(c)

Yes / No

(d)

Yes / No

46°90°

90°

39°51°

123°



2. On the following diagram, four paths are shown from point O to the line ST.

(a) Measure the following distances and angles.

OA = __________ mm OAT = __________

OB = __________ mm OBT = __________

OC = __________ mm OCT = __________

OD = __________ mm ODT = __________

(b) (i) Which is the shortest path? __________

(ii) What angle does the shortest path make with the line ST? __________

(c) Complete the following.

The shortest distance between a point and a straight line is the one whose path is

p__________________ to the line.

O

A B C DS T




Parallel lines are always the same distance apart. That is, they never intersect.

Example Which of these pairs of lines are parallel?

I. II. III.

Solution I and III show lines that are parallel.

1. The following two diagrams show pairs of lines PQ and RS. In both diagrams, the length of ACis equal to 2 cm.

(a) In both diagrams above, measure the length of BD to the nearest centimetre.

Length of BD in diagram 1 is _________.

Length of BD in diagram 2 is _________.

(b) (i) In which diagram are the lines PQ and RS parallel? _________________

(ii) How do you know this is true?

________________________________________________________________

________________________________________________________________

They look like train tracks.

A B

C D

P Q

R S

Diagram 1

A

B

C D

P

Q

R S

Diagram 2



2. Draw a line which is parallel to line AB and which also passes through the point N.


Arrowheads are used in the middle of lines to show that they are parallel.

1. Examine the following diagram and answer the questions below.

Name a line:

(a) parallel to AB __________

(b) parallel to TU __________

(c) perpendicular to AB __________

(d) perpendicular to PQ . __________


B

C

E

P

Q

R

S

A

D

F

T

U

A

B

N

Note how they could be single

arrowheads, double

arrowheads etc.



6. Angles and parallel linesWhen you complete this section you should be able to:

identify angle relationships within parallel lines.

Key words

parallel corresponding alternate co-interior

Warm-up 6

1. 97.09 10 = _________

2. 29 – 13 = _________

3. The temperature is minus 5 degrees.

How much will it need to increase to get to 8 degrees? _________

4.5 3

4 4 _________

5. 1

824 = _________

6. 67 800 mg = _________ g

7. 9 – 4 + 5 = _________

8. Write 1.125 as a percentage. __________

9. 1 2

3 35, 4 , 3 , ________

10. Determine the size of the missing angle.

_________ ?32°

0

5

10

20

°C



Focus problem 6

The following shows a set of parallel lines intersected by a transversal.

1. What part of the diagram tell you that the lines are parallel? ______________________

2. Use your own words to define ‘transversal’. ___________________________________

_______________________________________________________________________

_______________________________________________________________________

3. On the diagram above:

(a) Mark each of the acute angles with an arc and place the letter A within it.

(b) Mark each of the obtuse angles with an arc and place the letter O within it.

4. One of the acute angles measures 45. Without measuring the angles, answer the following:

(a) What do you think the other acute angles measure? __________________________

(b) What do you think the other obtuse angles measure? _________________________

5. Use your protractor and measure the angles to see if you are correct.

6. Explain why your measurements might not be the same as your answers for part 4?

_______________________________________________________________________

_______________________________________________________________________


The transversal is the line that cuts through

the pair of lines.




Corresponding angles Angles which are in similar positions on the parallel lines are called corresponding angles.

Corresponding angles are congruent.

Example Determine if the marked angles are corresponding for each of the following.

(a) (b) (c)

Solution (a) Yes

(b) No

(c) No

1. Mark the corresponding angle to the one already marked in each of the following.(a) (b) (c)

2. Find the size of the corresponding angle.

a = ___________

*

*

a°

120°

*

*#

#

#

#




Alternate angles Alternate angles are found between the parallel lines and on opposite (alternate) sides of the transversal.

Alternate angles are congruent.

Example Determine if the marked angles are alternate for each of the following.

(a) (b) (c)

Solution (a) No

(b) No

(c) Yes

1. Mark the alternate angle to the one already marked for each of the following.(a) (b) (c)

2. Find the size of the alternate angle.

a= ___________

*

*

#

#

a°

60°




Co-interior angles Co-interior angles are between the parallel lines and on the same side of the transversal.

Co-interior angles add up to 180.

Example Determine if the marked angles are co-interior for each of the following.

(a) (b) (c)

Solution (a) No

(b) Yes

(c) No

1. Mark the co-interior angle to the one already marked for each of the following.(a) (b) (c)

2. Find the size of the co-interior angle.

a= ___________

#

*

a°

120°

*

#

That means co-interior angles must also be

supplementary angles.



3. In each of the diagrams below, state whether the angles a and b are alternate, corresponding,co-interior, vertically opposite or none of these.

(a)

_________________________

(b)

_________________________

(c)

_________________________

(d)

_________________________

(e)

_________________________


b

a

b

a

b

a

ba

b

a



7. Using angle relationshipsWhen you complete this section you should be able to:

use angle relationships to determine unknown angles.

Warm-up 7

1. 67 10 = _________

2. 16 4 = _________

3. The temperature is 3 degrees.

How much will it need to decrease to get to minus 8 degrees?

_________

4.5 1

5 2 _________

5. 3

816 = _________

6. 20 mL = ________ L

7. 16 – (4 + 5) = _________

8. Find 10% of $75. _________

9. Describe the rule for the following pattern. 100, 81, 64, 49, 25, …

______________________________________________________

______________________________________________________

10.

At what point is the truck?

_________

x1 2 3 4 5

y

12345

0

5

10

20

°C



Focus problem 7

Dr Optical is researching a new glass-like material. As part of the research, he is looking at refraction of light through the material.

Light bends as it enters different material.

The sides of the material are parallel.

Help Dr Optical in determining the size of the angle labelled r.

Optical illusions

Our minds are powerful devices but even so, they can still be tricked. If you have some spare time and access to the internet, search for some more examples of optical illusions.


Light source

Material

65°

50°

r



Skills development 7

Example Without measuring, determine the unknown angles and explain how you found each one.

Solution a = 120 Vertically opposite angles are congruent.

a = 120

b = 120 Corresponding angles are congruent.

b = a

= 120

c = 60 Supplementary angles add to 180 (together b and c form a straight angle).

c = 180 – 120

= 60

1. Determine the size of the unknown angle. Give a reason.

(a)m = __________

Reason: _____________________________________________________________

(b) f = __________

Reason: _____________________________________________________________

a°

120°

c°b°

m°60°

f°37°



(c) n = __________

Reason: _____________________________________________________________

(d) q = __________

Reason: _____________________________________________________________

(e) e = __________

Reason: _____________________________________________________________

(f) a = __________

Reason: _____________________________________________________________

(g) b = __________

Reason: _____________________________________________________________

n°

q°160°

40°e°

a°45°

b°

121°



2. The following shows a transversal cutting through a pair of parallel lines. The angle markedwith a 1 is the same size as the angle marked with a 3.

What other two angles are equal in size to angle 1? ____________________

3. Determine the size of the unknown angle.

(a)

b = __________

(b)

y = __________

(c)

e = __________

(d)

h = __________

1 234

5 678

50°

e°

140°

y°

72°

b°

h°

114°



4. In each of the following, the two lines appear to be parallel, but are they actually parallel? Tickthe appropriate box and give a reason for your answer.

(a)

Yes

No

Not enough information

Reason: _____________________________________________________________

____________________________________________________________________

(b)

Yes

No


Reason: _____________________________________________________________

____________________________________________________________________

(c)

Yes

No


Reason: _____________________________________________________________

____________________________________________________________________

(d)

Yes

No


Reason: _____________________________________________________________

____________________________________________________________________

45°

135°

35°37°

130°

50°

63° 63°



5. Through how many degrees must AB be turned to make it parallel to CD?

___________________


45°

140°

AB

C D



8. Summary The size of an angle is the measure of turn or rotation. When two or more lines, line segments or rays intersect, angles are formed about the point of

intersection. This point is known as the vertex.

Angles are measured in degrees and can be classified according to their size.– An acute angle is between 0 and 90.– A right angle is exactly 90.– An obtuse angle is between 90 and 180.– A straight angle is exactly 180.– A reflex angle is between 180 and 360.– A revolution is exactly 360.

Adjacent angles are next to one another. They have a common arm (line, line segment or ray)and a common vertex.

Vertically opposite angles are formed when two lines intersect.

.

Vertically opposite angles are congruent

Complementary angles add up to 90.

Supplementary angles add up to 180.

Vertex

D

AB

C

common armcommon vertex



When two lines intersect at right angles (90°), the lines are perpendicular.

Parallel lines are always the same distance apart. That is, they never intersect.

Angles which are in similar positions on the parallel lines are called corresponding angles.

Corresponding angles are congruent.

Alternate angles are found between the parallel lines and on opposite (alternate) sides of thetransversal.

Alternate angles are congruent.

Co-interior angles are between the parallel lines and on the same side of the transversal.

Co-interior angles add up to 180.

#

*

*

#

*

*

#

#

*

*

#

#*

*

#

#



9. Review tasksThe following tasks will assist you to consolidate your learning and understanding of the concepts introduced in this resource, and assist you to prepare for assessments.

Task A

Name: _____________________________ Suggested time: 35 minutes

Actual time taken: __________

Instructions

Complete this work on your own.

You may use a calculator, but show how you got your answer.

Attempt every question. Take as long as you need and record the time in the space provided above after you have finished.

1. PRT is marked with an arc. Mark the angle that is vertically opposite to it.

2. Add another dot on each of the following diagrams so that the pair of angles are correctlydescribed.

(a) corresponding angles (b) alternate angles

(c) co-interior angles (d) vertically opposite angles

P

Q

S

T R



3. Complete the following statements.

(a) Complementary angles add to ________ degrees.

(b) Supplementary angles add to ________ degrees.

4. Without measuring, determine the sizes of the unknown angles. Give reasons.

(a) a = __________

Reason: _____________________________________________________________

____________________________________________________________________

(b) b = __________

Reason: _____________________________________________________________

____________________________________________________________________

(c) c = __________

Reason: _____________________________________________________________

____________________________________________________________________

(d) d = __________

Reason: _____________________________________________________________

____________________________________________________________________

a°108°

d°

b°37°

c°

126°



5. Without measuring, determine the size of the unknown angle for each of the following.(a)

a = __________

(b)

b = __________

(c)

c = __________

(d)

d = __________

6. Without measuring, determine the size of the unknown angle.

(a) (b)

a = _________ b = __________

b°53°

51°d°

b°39°

a°

121°

c°78°

a°

102°



7. Determine which of the following pairs of lines are perpendicular, parallel or neither. Give your reasons.

(a)

Reason: _____________________________________________________________

____________________________________________________________________

(b)

Reason: _____________________________________________________________

____________________________________________________________________

(c)

Reason: _____________________________________________________________

____________________________________________________________________

(d)

Reason: _____________________________________________________________

____________________________________________________________________

89°

101°

81°

81°

68°

112°

55°35°



Task B

Name: _____________________________ Suggested time: 55 minutes

Actual time taken: __________

Instructions

Complete this work on your own.

You may use a calculator, but show how you got your answer.

Attempt every question. Take as long as you need and record the time in the space provided above after you have finished.

The following is the Western Australian flag.

Note the Union Jack that is located in the top left corner. In particular, note the parallel and perpendicular lines of the Union Jack. The Union Jack is the flag for the United Kingdom.

1. Redraw the Union Jack in the space below.

2. From your diagram, only choose one angle to measure. Record its measurement on yourdiagram.

3. Without measuring, find the size of three other angles. Record these below and explain howyour determined them. (Label each angle on your diagram.)

Angle 1: ________ ____________________________________________________

Angle 2: ________ ____________________________________________________

Angle 3: ________ ____________________________________________________



4. Research other flags of the world.

(a) List five flags that feature perpendicular lines.

_________________________

_________________________

_________________________

_________________________

_________________________

(b) List five flags that feature parallel lines.

_________________________

_________________________

_________________________

_________________________

_________________________

5. In the space below, create your own flag using at least one set of parallel lines and one set ofperpendicular lines.



Self-evaluation task

Please complete the following.

How well did you manage your own learning using this resource?

Always Usually Rarely Not sure

Each section took approximately 45 minutes to complete.

I needed extra help.

I marked and corrected my work at the end of each section.

I made the journal entries and summaries when asked.

I have kept to my work schedule.

How much mathematics have you learnt using this resource?

Always Usually Rarely Not sure

Understanding I understand the difference between parallel and perpendicular lines.

I can identify angles formed by a transversal crossing a pair of lines.

Fluency

I can identify angle relationships within parallel lines.

I can use angle relationships to determine unknown angles.

Problem Solving I can solve problems involving angles and their relationships.

I can choose the correct relationship in determining the unknown angles.

Reasoning I can explain why lines are parallel or perpendicular based on angle relationships.



Write a list of topics for which you need additional assistance. Discuss these with your teacher.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________



Solutions 1. Angle naming conventions

Solutions to Warm-up 1

1. The factors of 15 are 1, 3, 5 and 15.

2. 18

3. (-5)

4. 2

3

5. 15

6. 11.7

7. 16.8

8. 7

8

9. 88

10. P(even) = 4 1

8 2

Solutions to Review 1.1

1. (a) CD

(b) QR

(c) AB

2. Solutions may vary.

S T

Note that the convention is to label alphabetically. However, rays have a slightly different

convention, as previously stated. Label rays using the point, from which it starts, as the first letter.



Solutions to Review 1.2

1. Solutions are as shown, with the larger angle ticked for each pair.

(a)

(b)

(c)

2.

3. Solutions may vary. Please check your answers with your teacher.An example of each angle is shown below.

(a) Smaller than angle A (b) Larger than angle A

arc

arm

vertexarm



Solution to Focus problem 1

What you were asked to do was to locate the angle that Mai is describing. The angle Mai is describing is shown below.

However, note the angle could be the larger angle as shown.

In general, the angle referred to is the smaller angle unless otherwise stated.

Solutions to Skills development 1.1

1. EDF , FDE , D

2. (a) CBD or DBC(b) EHI or IHE(c) MON , NOM or O(d) MON , NOM or O

3. (a)

(b)


1. ADB or BDA

2. d

CA

B

D

CA

B

D

E

H

T

KB

KBT

H

T

KB

KBT

You might have noticed that the angles in (c) and (d) are labelled

the same way. To distinguish between the two, sometimes the word reflex or an abbreviation is used in front of the reflex angle. For example, it might be labelled

as reflex MON , ref O or sometimes just r NOM .



2. Classifying angles


1. 32. 73. (-4) degrees

4. 1 2

3 6

5. 7.56. 97. 26.348. 0.379. 3.210. 83

Solutions to Review 2

1. Please check your estimates with another student or with your teacher. The correct sizes aregiven below.

(a) 35(b) 160(c) 135(d) 110(e) 42


What you were asked to do was to classify the angles using their sizes.

Angle A is less than 90. It is an acute angle.

Angle B is greater than 90. It is an obtuse angle.

Solutions to Skills development 2

1. (a) 90(b) 180(c) 90(d) 180(e) 360(f) 360

2. (a) Obtuse(b) Acute(c) Straight(d) Revolution(e) Reflex(f) Right(g) Obtuse



3. Adjacent and vertically opposite


1. 9 should be circled2. 243. (-2)4.

5. 106. 38 – 21 = 177. 24.638. 16.666…% or 16.6%

9. 4

66

10. (0, 2)


What you were asked to do was to determine the unknown angles given the one angle, as shown.

If you were to measure the angles above, you will notice that the angle opposite the 62 is the same size. The other two angles are equal too, but different to the 62. You will be able to use this information to find all the missing angles.

For your information, the other angles are both 118. Can you work out why?


1.

2. (a) b, e

(b) a, c

(c) b, d

(d) c, e

(e) d, a

0 1

34

E

HF G

Obtuse and adjacent to EHF

62°




1.

2. Solutions will vary. Please check you definition with another student or your teacher. Thefollowing is a possible answer: congruent means to have the same shape and size.

3. (a) (i) a° is the same size as b°

(ii) x° is the same size as y°

(b) Solutions may vary. A possible answer is yes, it will always occur, as vertically oppositeangles are equal in size. When you rotate one of the lines, it changes both pairs of angles bythe same amount.

4. (a) Yes

(b) Yes

(c) Yes

5. Equal

6. (a) (i) x = 140

(ii) y = 40

(b) (i) m = 65

(ii) n = 115

(c) (i) a = 60

(ii) b = 90 (Note how the right angle was marked in the question.)

(iii) c = 30

4. Complementary and supplementary


1. 3

2. 9

3. 4 degrees

4.5

6

5. 12

P

Q

S

T R

vertically oppositePRT



6. 201

7. 0.81

8.1

3

9. 198

10. P(composite number) = 2 1

or6 3


What you were asked to do was to find the adjacent angle to 50 as marked with the arc.

The two angles form a straight angle when combined. That is, the angles add to 180. We can use this knowledge to find the missing angle.

50 + ? = 180 ? = 180 – 50

= 130

Therefore, the missing angle is 130. We can also find unknown angles given that they add to 90 using a similar method.


1. (a)

Complementary (25 + 65 = 90)

(b)


(c)

Not complementary (64 + 46 ≠ 90)

(c) 24 and 56


(d)


(e)


50°

64°46°

71°19°30°

70°

25°

65° 47°

43°



2. (a) y = 34

(b) x = 61

3. (a) 12

(b) 81

(c) 67


1. (a)

Supplementary (60 + 120 = 180)

(b)


(c) 124 and 56


(d)

Not supplementary (27 + 126 ≠ 180)

2. 180

3. (a) 150

(b) 55

(c) 77

4. y° = 95

5. (a) The square-marking of the angle shows that it is a right angle or 90.

(b) a° = 90

b° = 90

c° = 90

5. Perpendicular and parallel lines


1. 6 = 2 × 3

2. 41

3. 2

47°133°

27° 126°

60°120°



4. 14 4 2

10 10 5or 1 or 1

5. 18

6. 708 m

7. 5

8. 4.166... or 4.16

9. 50

10. Circle


What you were asked to do was to describe the meaning of the terms perpendicular and parallel. You might like to check your definitions with another student or your teacher as they may vary slightly.

Perpendicular means to intersect at or form a right angle (90). These two lines are perpendicular as indicated by the square-marked angle.

Parallel means to be equal distance apart and never intersecting. These two lines are parallel as indicated by the arrows in the middle of the lines.


1. (a) Yes(b) Yes(c) No(d) Yes

2. Solutions may vary slightly. Please check your solutions with your teacher.The following is an example of what you might expect to get.

(a) OA = 81 mm OAT = 30 OB = 57 mm OBT = 45 OC = 45 mm OCT = 63 OD = 40 mm ODT = 90

(b) (i) OD(ii) 90 (right angle)

(c) Perpendicular




1. (a) 2 cm.1 cm.

(b) (i) Diagram 1(ii) They are parallel because AC and BD are equal in size.

2. Check your solution with another student or your teacher.The following shows the line EF which is parallel with the line AB and passesthrough N.


1. (a) CD

(b) PQ

(c) RS

(d) EF

6. Angles and parallel lines


1. 970.9

2. 16

3. 13 degrees

4.2 1

or4 2

5. 3

6. 67.8 g

7. 10

8. 112.5%

9. 3

10. 148

A

B

N

EF



Solutions to Review 6

Solutions may vary slightly. Please check your solutions with another student or your teacher.

1. The arrowhead in the middle of the two lines

2. A transversal is a line that passes through a set of lines.

3. See the diagram below.

4. (a) 45

(b) 135

5. No solution required.

6. Inaccuracies with measuring and/or measuring device.


1. (a) (b) (c)

2. 120


1. (a) (b) (c)

2. 60

AA

AA

OO

OO




1. In each of the following, mark the co-interior angle given the one already marked.(a) (b) (c)

2. 60

3. (a) Co-interior

(b) Not one of the given relationships

(c) Alternate

(d) Corresponding

(e) Vertically opposite

7. Using angle relationships


1. 6.7

2. 64

3. -11 degrees

4.1

2

5. 6

6. 0.02 L

7. 7

8. $7.50

9. Starting from 100, it is the square numbers in reverse order.

That is, 10 × 10 = 100, 9 × 9 = 81, 8 × 8 = 64, etc

10. (1, 5)




What you were asked to do was to determine the missing angle. If you look carefully, you might be able to recognise a set of parallel lines intersected by a traversal.

We can use the relationships in parallel lines and other angle relationships to help find any unknown angles.

For example, the angle r is alternate to 50. That is, angle r is 50.

Solutions to Skills development 7

1. (a) m = 120 (supplementary angles)

(b) f = 53 (complementary angles)

(c) n = 90 (supplementary angles and the right angle is indicated)

(d) q = 160 (vertically opposite angles)

(e) e = 40 (vertically opposite angles)

(f) a = 45 (vertically opposite angles)

(g) b = 121 (alternate angles)

2. The angles marked 5 and 7 are the same size as the angle marked with a 1.

3. (a) b = 72

(b) y = 140

65°

50°

r

r

50°



(c) e = 130

(d) h = 114

4. Reasons may vary. Please check your solutions with your teacher.(a) Yes – the 45 is supplementary to 135 and it meets the condition of corresponding angles.

(b) No – corresponding angles are not equal.

(c) Yes – the 50 is supplementary to 130 and it meets the condition of corresponding angles.

(d) Not enough information

5. It must be turned 5 in an anticlockwise direction or 355 in a clockwise direction.

45°

135°

135°45°

45°

45°

135°

135°

130°

50°

50°

50°

50°130°

130°130°

63° 63°117°

117° The size of the missing angles will only match the other angles if the two lines are parallel.



Solutions to Review tasks

Solutions to Task A

1.

2. (a) corresponding angles (b) alternate angles

(c) co-interior angles (d) vertically opposite angles

3. (a) 90

(b) 180

4. (a) a = 72 (supplementary angles)

(b) b = 37 (vertically opposite angles)

(c) c = 126 (alternate angles)

(d) d = 90 (supplementary with the right angle marked)

5. (a) a = 121

(b) b = 37

(c) c = 102

(d) d = 51

6. (a) a = 78

(b) b = 51

7. (a) Perpendicular – adjacent angles add to 90.

(b) Neither – co-interior angles do not meet the condition of adding to 180.

P

Q

S

T R



(c) Parallel – corresponding angles are equal.

(d) Parallel – the vertically opposite angle is 68, which then meets the condition ofco-interior angles adding to 180.

Solutions to Task B

1. Simplistically, the Union Jack can be created using sets of parallel and perpendicular lines, asshown below.

2. The size of the angle will depend on the accuracy of the diagram and the angle chosen.For example, the following shows one of the angles.

3. Solutions will vary depending on part 2 and the angles chosen. For example, using the anglegiven in the part 2 solutions, the following angles can be found.

Angle 1: 63 – it is vertically opposite.

Angle 2: 27 – it is complementary.

Angle 3: 27 – it is vertically opposite to angle 2.

4. Solutions will vary. A list of five flags for each category follow but it is not exhaustive.

(a) Flags featuring perpendicular lines include the Beninese, Danish, Georgian, Norwegian andTongan flag.

(b) Flags featuring parallel lines include the Austrian, Belgian, French, Polish and Russianflags.

5. Solutions will vary. Students should show a flag with at least one set of parallel lines and one setof perpendicular lines.

63°

63°Angle 2

Angle 1

Angle 3

MATHSAC027 ANGLES

ISBN: 9780730744474

australian curriculum mathematics year 7 · review 1.1 lines, rays and segments a line continues in...

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