Chapter02 Angles

Geometry

Angles

2

Whether you are playing snooker or soccer, building houses or bridges, or designing logos or computer graphics, you need to know all about angles. Look around you for a moment—you will see angles

■ name and classify different types of angles and lines

■ revise protractor skills for measuring and drawing angles

■ recognise and use notations for angles, parallel lines and perpendicular

lines

■ discover different angle facts and use them to solve geometry problems

■ identify and measure pairs of alternate, corresponding and co-interior

angles for two lines cut by a transversal

■ discover and use the properties of alternate, corresponding and

co-interior angles for two parallel lines cut by a transversal.

■ vertex The corner or point of an angle.

■ protractor An instrument for measuring the size of an angle. ■ complementary angles Two angles that add to 90°.

■ supplementary angles Two angles that add to 180°.

■ parallel lines Lines that point in the same direction and do not intersect. ■ perpendicular lines Lines that intersect at right angles.

■ transversal A line that cuts across two or more other lines.

The Earth takes one year to make a complete revolution around the Sun. How long does it take to travel an angle of 30°?

In this chapter you will:

Wordbank

1 How many degrees are there in: a a quarter turn (right angle)? b a half turn (straight angle)? c a three-quarter turn?

2 In this diagram, each gap represents 1° of angle size.

What is the angle, in degrees, between the lines labelled:

a A and C? b A and D? c B and C?

d C and F? e A and F? f B and G?

g D and G? h E and H? i D and I?

j C and J? k B and E? l E and J?

3 In the diagram in Question 2, find one pair of labelled lines which have a 19° angle between them.

4 In the diagram in Question 2, find two pairs of labelled lines which have a 90° angle between them.

5 In the diagram in Question 2, find the pairs of labelled lines which have the following angles between them:

a 7° b 8° c 13° d 28°

e 50° f 89° g 95° h 114°

6 The word ‘degree’ has many meanings. Find four non-mathematical meanings for the word. A B C D E F G H I J

Start up

Naming angles

An angle is a description of the size of a turn or rotation. It is drawn with two arms which meet at a vertex. Angles are normally marked with a curved line called an arc. This shows the size of the turn. The angle marked in this diagram can be written as:

• ∠G or

• ∠PGH or ∠HGP • P H or H P

7 Decide whether each of these angles is:

i acute ii obtuse iii reflex

a b c d e f g h i j k l m n o Skillsheet 2-03 Types of angles G P H vertex arm arc

The middle letter is always the letter that labels the vertex of the angle.

{

Gˆ

Example 1

Name the angle marked with in each of these diagrams.

Solution

a ∠Y or ∠XYZ or ∠ZYX b ∠PQS or ∠SQP

Note: We cannot name this ∠Q because it is not clear which angle that means. There are three different angles whose vertex is ∠Q. They are ∠PQS, ∠SQR and ∠PQR.

X Y Z P Q S R a b

1 Name each of these angles in two different ways:

2 Name the angle marked with in each of these diagrams:

a P b c Q K _O R C G V _E A G T P Q D R C D d e f D B C A _N M Q P P T S R Q F E H C B A D Z W Y a b c d e f X E G

Exercise 2-01

Comparing angle size

Can you tell which of these is larger?

Angle A Angle B

If you can tell these apart, your eyes can detect a difference of two degrees. Angle B is 2° larger than Angle A.

3 Draw each of these angles, labelling them correctly:

a ∠POT b ∠TAF c ∠AFE d ∠H

4 a There are 13 different angles inside this diagram. Name them all.

b What type of angle is ∠NCY?

5 Name the angles marked and in each of the following diagrams:

6 Angles AMP and PMN share a common arm, PM. They also share a common vertex, M. Angles that are next to each other in this way are called adjacent angles.

Name a pair of adjacent angles for each diagram in Question 5. N A Y D C x a b c d e f A D C B R S P Q Q R P M N Z Y X W H F D E G A B C G H I E F x x x x x x N M A P arm

1 List the angles in each of the given sets in order, from smallest to largest, without using a protractor.

a These angles vary by 5°.

b These angles vary by 3°.

c These angles vary by 2°.

d These angles vary by 1°.

e These angles vary by °.

f These 10 (i to x) angles will provide a harder challenge:

i ii iii i ii iii i ii iii i ii iii 1 2 ---i ii iii i ii iii iv v vi

Exercise 2-02

The protractor

Measuring angles

A protractor is an instrument used to measure angles.

vii viii ix x

2 List the following angles in order, from smallest to largest, without using a protractor.

a b c

f e

d

Worksheet 2-03 Make your own

protractor Worksheet 2-02 360° scale Worksheet 2-04 A page of protractors Centre mark 90 100 110 120 130 140 150 160 170 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 10 0 100 110 120 130 140 150 160 170 180 Base line Outside scale Inside scale

Example 2

1 Measure angle AOB.

Solution

• Line up OB with the base line of the protractor. • Place the centre mark over the vertex, O. • The angle is smaller than 90°.

• Use the inside scale, counting from 0°.

Angle AOB = 54°.

2 Measure ∠PMQ.

Solution

• Line up QM with the base line of the protractor. • Place the centre mark over the vertex, M. • The angle is greater than 90°.

• Use the outside scale, counting from 0°. ∠PMQ = 155°. B A O 90 100 110 120 130 140 150 160 170 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 10 0 100 110 120 130 140 150 160 170 180 B A O M Q P 90 100 110 120 130 140 150 160 170 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 10 0 100 110 120 130 140 150 160 170 180 M Q P

3 Measure ∠TEX.

Solution

• Line up TE with the base line of the protractor. • Place the centre mark over the vertex E. • ∠TEX is bigger than 90°.

• Use the inside scale. ∠TEX = 134°

Measure the reflex angle ∠GHK.

Solution

• Actually measure the obtuse angle ∠GHK first (140°). • Subtract 140° from 360°. 360 − 140 = 220 Reflex ∠GHK = 220° X E T 90 100 110 120 130 140 150 160 170 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 10 0 100 110 120 130 140 150 160 170 180 X E T

Example 3

1 Find the size of each of these angles:

2 Estimate the size of each of these angles. Name each angle and use a protractor to measure the angles accurately.

90 100 110 120 130 140 150 160 17 0 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 1 0 0 100 110 120 130 140 150 160 170 180 E T 90 100 110 120 130 140 150 160 170 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 1 0 0 100 110 120 130 140 150 160 170 180 B A O O a b 90 100 110 120 130 140 150 160 17 0 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 1₀ 0 100 110 120 130 140 150 160 170 180 90 100 110 120 130 140 150 160 170 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 1₀ 0 100 110 120 130 140 150 160 170 180 D N O P O M c d 90 100 110 120 130 140 150 160 17 0 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 10 0 100 110 120 130 140 150 160 170 180 G U Y _{90 100 110} 120 130 140 150 160 17 0 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 1₀ 0 100 110 120 130 140 150 160 170 180 L A F 90 100 110 120 130 140 150 160 17 0 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 1₀ 0 100 110 120 130 140 150 160 170 180 I U R 90 100 110 120 130 140 150 160 170 180 80 70 60 50 40 30 20 10 0 90 80 70 60 50 40 30 20 10 0 100 110 120 130 140 150 160 170 180 H B K e f g h A B O P Q a b D

Exercise 2-03

N M A Y X P S Z X Y T c d e f g M N L i j k l h G D A M G E C A B Z Q F D P B F

3 Estimate the size of each of these angles. Name each angle and use a protractor to measure the angles accurately.

4 Measure the angles marked with and on each of these diagrams.

a b c d C B A N M L X Z Y G K H x a b c d x x x x Example 3

e f x x Skillsheet 2-01 Starting Geometer’s Sketchpad Skillsheet 2-02 Starting Cabri Geometry

Estimating angles

1 Using a drawing package and estimation skills, draw and label each of the following angles.

a ∠ABC = 45° b ∠DEF = 30° c ∠GHI = 60°

d ∠JKL = 90° e ∠MNO = 120° f ∠PQR = 155°

2 Print out the six angles that you have drawn. Measure the angles with a protractor. 3 What was the error each time between the angle you drew and the actual angle

requested?

Using technology

Why 360 degrees?

Why are there 90° in a right angle and 360° in a revolution? Why do we use such strange numbers instead of more conventional numbers like 10 and 100?

The reason is that, in 2000 BC, the ancient Babylonians used a base 60 system of

numbers. They used a base 60 number system because:

• 60 is a rounder, more convenient number which has more factors than 10. You can divide 60 by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.

• 6 × 60 = 360, which was the Babylonian approximation of the number of days in a year.

They defined a revolution as being 360° so that, each day, the Earth would travel 1°

around the Sun. A right angle, being a quarter-revolution, thus became 360° ÷ 4 = 90°.

Some people who prefer a base 10 system of measurement use grads instead of degrees to measure angles. With this system, a right angle is 100 grads and a revolution is 400 grads. Find out more information about grads, including the exact relationship between degrees and grads.

Drawing angles

You can also use your protractor to draw angles.

Example 4

Use a protractor to draw angle KPM which measures 76°.

Solution

• Draw a line with endpoints P and M.

• Line up the base line of the protractor over PM. Place the centre mark over P. Follow the inside scale around on the protractor, from 0° to 76°. Mark this point.

• Draw a line from P through this mark. Label the end of this line K.

You have now drawn angle KPM, measuring 76°. M P 90 100 110 120 130 140 150 160 170 180 80 70 60 50 40 30 20 10 0 90 80 ₇₀ 60 50 40 30 20 10 0 100 110 120 130 140 150 160 170 180 P M

choose scale with 0° near M mark 76° P M K line ruled from P through mark at 76°

1 Accurately draw these angles, using your protractor:

a 35° b 115° c 150° d 40°

e 15° f 170° g 117° h 200°

2 Use your protractor to accurately draw and label these angles:

a ∠DRE = 65° b ∠BGH = 145° c ∠GRT = 32°

d ∠ABC = 45° e ∠SAQ = 110° f ∠NMH = 265°

g ∠KLY = 28° h ∠LMN = 180° i ∠LKY = 90°

3 You can use a geometry program, such as Cabri Geometry or Geometer’s Sketchpad, to draw accurate angles. The accompanying activity shows you how to make a protractor.

Exercise 2-04

Angle geometry

Classifying angles

Angles may be classified according to their size:

Angle Type Description

acute less than 90°

right 90° (quarter turn)

Note that a right angle is marked with a box symbol.

obtuse greater than 90° but less than 180°

straight 180° (half turn)

reflex greater than 180° but less than 360°

revolution 360° (complete turn)

Worksheet 2-06 Angle cards Skillsheet 2-03 Types of angles

1 Draw two different examples of:

a an acute angle b an obtuse angle c a right angle d a reflex angle e a straight angle f a revolution 2 Classify each of the following angles:

a 37° b 107° c 252° d 195° e 79° f 180° g 163° h 179° i 360° j 5° k 345° l 91° m 14° n 299° o 90° p 205° q 126° r 44°

Exercise 2-05

3 Decide whether each of these angles is acute, obtuse or reflex.

4 Copy and complete the information below each of these diagrams. Use your protractor to measure the angles.

a b

∠ABD = ∠XYZ =

∠CBD = ∠XZY =

∠ABD + ∠CBD = ∠XYZ + ∠XZY =

(The angles you measured are called complementary angles. They complement each other to form 90°.)

5 Look up ‘complement’ in a dictionary. Write one non-mathematical meaning you find. 6 What is the complement of:

a 30°? b 70°? c 25°? d 38°? e 89°? f 57°? g 42°? h 66°? i 11°? j 74°? k 1°? l 12°? A B C Z Y X D

Complementary angles add to 90°.

a b c d SkillBuilder 24-05 Complementary angles

7 Copy and complete the information below each of these diagrams. Use your protractor to measure the angles.

a b

∠ABD = ∠PQR =

∠CBD = ∠SRQ =

∠ABD + ∠CBD = ∠PQR + ∠SRQ =

(These pairs of angles are said to be supplementary. They supplement each other, together forming 180°.)

8 Look up ‘supplement’ in your dictionary. Write a non-mathematical meaning for it. 9 What is the supplement of:

a 18°? b 150°? c 35°?

d 125°? e 62°? f 87°?

g 111°? h 173°? i 54°?

j 132°? k 8°? l 91°?

10 a How many degrees are there in a complete turn or revolution? b Copy and complete the statements below each of these diagrams.

i ii

∠ADB = ∠AEB =

∠ADC = ∠BEC =

∠BDC = ∠CED =

∠ADB + ∠ADC + ∠BDC = ∠DEA =

∠AEB + ∠BEC + ∠CED + ∠DEA = (These angles all meet at a point.)

11 Use Cabri Geometry or Geometer’s Sketchpad to illustrate the meaning of as many angle words as you can.

D A B C P S Q R

Supplementary angles add to 180°.

A D C B D E A B C

Angles at a point (in a revolution) add to 360°.

SkillBuilder 24-04 Supplementary angles Geometry 2-02 Angle vocabulary

Vertically opposite angles

When two lines cross, four angles are created. • Which of these angles are equal?

• Can you prove it using supplementary angles?

12 Use the given information to find the size of the angle shown by the letter each time.

13 You can use a geometry program, such as Cabri Geometry or Geometer’s Sketchpad, to demonstrate the rules for supplementary angles and angles at a point. Use this link to go to the accompanying activity.

m° 160° q° 150° 170° 70° 62° 87° x° 120° y° 95° 25° 102° a° 135° 116° 22° d° 71° 55° 110° 105° w° 132° 123° f° 48° b a c d f e g h j i k l 30° 220° n° 152° k° 118° t° 47° 15° h° 303° Geometry 2-03 Revolutions and straight angles a° b° c° d°

Example 5

∠WKZ is vertically opposite and equal to ∠XKY. What angle is vertically opposite ∠ZKY?

Solution

∠WKX is vertically opposite ∠ZKY.

Note: Angles that are equal in size are marked on diagrams with the same type of arc or symbol.

W

Z Y

X

Vertically opposite angles are equal.

Example 6

Find the size of the angles shown by the letters in this diagram.

Solution

k = 130 m = 50

(since vertically opposite angles are equal).

130° 50°

k° m°

1 What angle is vertically opposite to:

a the angle marked a°? b the angle marked w°? c the angle marked c°?

d the angle marked h°? e the angle marked k°? f the angle marked m°?

2 Without measuring, find the size of the angle shown by the letter each time.

b° a° d° c° u° w° v° u° d° a° c° b° f ° e° h° g° k° d° i° h° p° n° m° l° 70° a° 110° m° 85° k° m ° 90° 135° x° 25° f ° a b c d e f

Exercise 2-06

Angle facts

Remembering these facts will help you complete the next exercise.

Types of angles Meaning Diagram

Adjacent angles Angles that share a common arm and a common vertex.

(∠ABD and ∠DBC are adjacent angles.)

Complementary angles

Two angles that add to 90°. (a + b = 90)

Supplementary angles

Two angles that add to 180°. (m + n = 180)

Vertically opposite angles

Formed when two straight lines cross. Vertically opposite angles are equal.

(a = c, b = d) Angles at a point Form a revolution and

add to 360°. (a + b + c = 360) w° _133° 29° n° 62° q° 163° t° h° g° 160° 20° r° s° q° 90° g h i j k l B D C A x a° b° m° n° c° b° a° d° a° c° b°

1 a If ∠TAF = 42°, what is the size of its complementary angle? b If ∠ZAB = 127°, what is the size of its supplementary angle? 2 Refer to the diagram on the right:

a Which angle is vertically opposite to ∠NDP? b Which angle is equal to ∠MDQ?

c Name two straight angles in the diagram.

d Name two different pairs of supplementary angles in the diagram.

3 Refer to the diagram on the right: a Name a pair of adjacent angles. b Name a pair of complementary angles.

c How do you know that the angles you named are complementary?

4 Calculate the size of the angle shown by the letter each time.

D P Q N M Q P R S 67° 23° a° 120° 70° a° 100° y° 45° m° 150° p° 19° m° _41° x° f° 15° 100° 100° 40° a° c b a f e d i h g

Exercise 2-07

Naming lines

A line is named using two points on the line. For example, this is the line AB.

When two lines cross, we say that they intersect. Two lines intersect at a point and form four angles between them.

For example, in this diagram, line DE intersects with line FG at point H. One of the angles formed is ∠FHE.

a° 32° b° 82° 135° y° d° d° f° e° e° 112° 48° l° k° j° 118° 75° y° x° _{85° 155°} p° y° x° 20° p° r q s t p o n m l k j 170° h° a° a° t° t° e° u e° e° B A D E G F H

Perpendicular lines

Lines that intersect at right angles are called perpendicular lines. For example, in this diagram, PQ is perpendicular to XY.

This is written as ‘PQ
XY’, where the
symbol stands for ‘is perpendicular to’.

Parallel lines

Lines that point in the same direction and never intersect are called parallel lines. Parallel lines are marked with identical arrowheads and are always the same distance apart. For example, in this diagram, MN is parallel to RS.

This is written as ‘MN II RS’, where the symbol II stands for ‘is parallel to’.

X P Q Y M R S N

indicates these lines are parallel

1 Name the six different lines in this diagram.

2 In this diagram, name two lines that: a are perpendicular

b are parallel c intersect?

3 Rewrite your answers to Question 2 parts a and b using the symbols for ‘is perpendicular to’ and ‘is parallel to’.

4 Draw and label correctly:

a line FG b line AB intersecting line CD at point E c line PQ parallel to line YZ d line JK perpendicular to line LM. 5 In your diagram for Question 4b, name two angles that are:

a adjacent b vertically opposite c supplementary.

A B C D G F E D C B A H

Exercise 2-08

Angles and parallel lines

A line that crosses two or more other lines is called a transversal. Transverse means ‘across’.

If a pair of parallel lines are crossed by a transversal, then special pairs of angles are formed: alternate angles, corresponding angles and co-interior angles. We shall identify these angles and discover their properties.

6 State all the examples of parallel lines, perpendicular lines and intersecting lines you can find in this photograph.

transversal transversal

Alternate angles

Alternate angles are on opposite sides of the transversal but between the parallel lines. They are marked with red dots on the diagram.

‘Alternate’ means ‘going back and forth’.

Draw a pair of parallel lines and mark the alternate angles as shown. Draw in the broken line and cut along it.

Rotate the two alternate angles and place them on top of each other. You should see they are the same.

The marked pairs of angles are alternate. Measure them and check that alternate angles are equal. (Remember: Equal angles are marked by the same symbol.)

Alternate angles on parallel lines are equal.

Pairs of alternate angles

x x

1 Which angle is alternate to the marked angle each time?

2 Copy each of these diagrams and mark in the alternate angle to the one shown each time.

a b c a° b° _c° g° _{f °} e° d° f ° g° e° c° b° d° a° g° f ° e° c° b° a° d° a b c

Exercise 2-09

Corresponding angles

Corresponding angles are on the same side of the transversal and are both either above or below the parallel lines. ‘Corresponding’ means ‘matching’.

3 Copy each of these diagrams and mark in a pair of alternate angles on each one.

a b c

4 Without the use of instruments, calculate the size of each angle shown by a letter:

a b c d e f g h i m° 110° 50° a° n° 80° 122° b° h° 20° n° m° p° 50° b° a° 40° 130° a° b° b° a° c° _44°

Corresponding angles on parallel lines are equal.

Pairs of corresponding angles

x x

We can prove that corresponding angles are equal using the following method:

a = b They are vertically opposite angles.

b = c They are alternate angles.

So a = c.

a° b°

c°

1 Which angle is corresponding to the marked angle each time?

a b c

2 Copy each diagram and mark the corresponding angle to the one shown each time.

a b c

3 Copy each of these diagrams and mark in a pair of corresponding angles on each one.

a b c

4 Calculate the size of each angle shown by a letter:

c° b°a° g° f ° e° d° f ° e° g° a°b° c° a° b° d° d° c° g° e° f ° d e f a° y° m° c° t° a° b° 120° 28° 108° 74° 63° 60° 50° a b c

Exercise 2-10

Co-interior angles

Co-interior angles are on the same side of the transversal but between the parallel lines. ‘Co-interior’ means ‘together inside’.

Measure the following pairs of angles and see if they really are supplementary.

We can also show that co-interior angles add to 180° using the following method:

a + b = 180 They are angles on a straight line.

a = c They are alternate angles.

So c + b = 180

5 Without measuring, find the size of the other seven angles in this diagram.

g h i m° 110° 105° n° c° y° 140° y° a° d° x° 105°

Co-interior angles on parallel lines are supplementary. They add to 180°.

Pairs of co-interior angles

x

x

a° b°

Example 7

1 Find the size of the angle marked a° in this diagram.

Solution

a + 80 = 180 Co-interior angles are supplementary.

a = 180 − 80 So a = 100

2 Find the size of the angle marked m° in this diagram.

m + 55 = 180 Co-interior angles are supplementary.

m= 180 − 55 So m= 125 a° 80° 55° m°

1 Which angle is co-interior with the marked angle each time?

a b c

2 Copy each of these diagrams and mark the angle that is co-interior with the marked angle each time.

a b c

3 Copy each of these diagrams and mark pairs of co-interior angles.

a° d° b° c° g° f° e° a°b°c° g° d° e° f° a° b°_c° d° e° f ° g° a b c

Exercise 2-11

4 Without the use of instruments, calculate the size of the angles shown by letters: d e a° 50° _m° 90° 75° b° 112° d° 68° m° 98° a° b° f f ° g° 130° k° j° 55° c° b° a° 51° g h i a b c Example 7

The Leaning Tower of Pisa

The Leaning Tower of Pisa, Italy, began leaning shortly after its construction commenced in 1173. In 1350, it was leaning at 2.5°, or 4 m, from the vertical. By 1990, its lean had grown to 5.5°, or 4.5 m, and was increasing at 1.2 mm per year. Architects estimated that the tower would have toppled over by the year 2020 so it was closed for 12 years to allow $25 million worth of engineering work to take place. When it reopened in 2001, its lean had been pushed back to 5° or 4.1 m, and it is now guaranteed to stay up for at least another 300 years. 1 Draw a scale diagram of the Leaning Tower

of Pisa given that its top is 55 metres above the ground.

2 Research how engineers prevented the tower from leaning further. Use the library or the Internet to conduct your research.

Just for the record

Summary

When parallel lines are crossed by a transversal: • alternate angles are equal

• corresponding angles are equal

• co-interior angles are supplementary (add to 180°). Worksheet

2-07 Matching

angles

1 In the diagram on the right, name the angle that is: a corresponding to ∠VWA

b alternate to ∠QXW c co-interior with ∠PWX d supplementary with ∠AWX e alternate to ∠SXV

f corresponding to ∠ZXS.

2 Without the use of instruments, find the size of each angle shown by a letter:

d e f g h i j k l Q A X Z W V P S p° 115° 71° t° 105° k° a b c 120° m° 70° 132° n° a° x° 28° 72° s° k° 85° p° 93° 81° y° 150° _w°

Exercise 2-12

m n o

3 Without measuring, find the size of all angles labelled with letters in these diagrams:

4 You can use Geometer’s Sketchpad or Cabri Geometry to show that the rules for parallel lines and traversals are always true. The instructions for this can be found in the

accompanying activity. 128° d° j° 66° q° 109° j k l g h i d e f m n o a b c b° 67° a° 133° j° k° l° m° n° p° 52° y° z° 42° 95° l° m° b° c° 45° 30° q° p° 75° 85° m° k° p° w° 63° k° 130° x° y° 55° 62° a° 72° b° n° p° m° 83° 132° g° 27° a° b° c° SkillBuilders 24-10 to 24-12 Combination figures Geometry 2-04 Angles and parallel lines

Finding parallel lines

We can use what we know about angles and parallel lines to show that two lines are parallel.

Example 8

1 Is AB parallel to CD in the diagram on the right?

Solution

∠AXY is alternate to ∠DYX. ∠AXY = ∠DYX = 75°

∴ AB II CD since a pair of alternate angles are equal.

(∴ means ‘therefore’)

2 Is MN parallel to PQ in the diagram on the right?

Solution

∠MXY is co-interior with ∠PYX. ∠MXY + ∠PYX = 110° + 80° = 190°

≠ 180°

Since co-interior angles do not add to 180°, MN is not parallel to PQ.

75° Y 75° X B D C A 80° 110° M P Y Q N X

1 In each diagram below, name a pair of alternate angles and use them to decide if AB is parallel to CD.

a b c

2 In each diagram below, name a pair of corresponding angles and use them to decide if AB is parallel to CD.

a b c

3 In each diagram below, name a pair of co-interior angles and use them to decide if AB is parallel to CD. a b c 64° 64° A B D C 100° A C D B 100° A C D B 32° 35° E F G H E F C A B D 79° 82° A C B D 63° 63° C D A 117° 110° B G E F E F G E F G A B D C A B C D 120° 60° 100° 85° A C B D 90° 90° E F E F E F

Exercise 2-13

4 For each diagram below, determine if line PQ is parallel to line MN. Explain your reasons. P A M C D Q N B 99° 81° N Q Y X P M E G I K M P F H J L Q N 87° 87° 78° 102° a b c 78° 78° P N M X Q 105° f K D M P C Q L A 65° 120° d P A M E D Q N B 80° 95° e N 65° B 80° C 95° 85° F 75° 75° Example 8

1 a Draw any triangle with angles of 70° and 55°. b Draw any parallelogram with angles of 50° and 130°.

c Draw any four-sided shape with angles of 45°, 160°, 70° and 85°. 2 a Draw any triangle and measure the sizes of all three angles.

b What is the sum of the angles in any triangle?

c Draw any quadrilateral and measure the sizes of all four angles. d What is the sum of the angles in any quadrilateral?

3 How many degrees does the Earth spin on its axis in:

a one day? b one hour? c 8 hours? d 10 minutes?

4 Work out which direction (left, right, front or behind) you would be facing after making each of these series of turns.

a Right 80°, right 240°, left 90°, right 40° b Left 140°, left 140°, left 140°, right 60° c Right 200°, left 70°, right 40°, right 10° d Left 240°, right 190°, right 100°, left 50°

5 Find the size of each angle shown with a letter. Give reasons for your answers. a b c 51° m° 62° x° y° 125° a° 82° 40° m° y° 35° 250° c° 80° 145° k° 50° x° 35° 120° m° 45° _20° 95° k° d e f g h i

Language of maths

acute adjacent alternate arc

arm co-interior complementary corresponding

degree intersecting line obtuse

revolution right angle scale straight angle

supplementary transversal vertex vertically opposite 1 Name the two parts of an angle.

2 Find out what the word ‘acute’ means when referring to a disease, for example acute appendicitis.

3 What is the difference between ‘complementary’ and ‘complimentary’? 4 When something happens that dramatically changes the way we think or do

things, it is called ‘revolutionary’. Why do you think this is so? 5 Write the mathematical symbol for:

a parallel b perpendicular

6 Mr Transversal visits his parents on alternate days. What does this mean? How is it similar to the mathematical meaning of ‘alternate’?

Worksheet 2-07 Matching angles Worksheet 2-08 Angles crossword

Topic overview

• Give three examples of where angles are used. • Do you think this chapter is very useful to you? Why? • How confident do you feel in working with angles?

• Is there anything you did not understand about angles? Ask a friend or your teacher for help. • The diagram below provides a summary of this section of work. Copy it into your workbook

and complete it, using colour, pictures and key words to make your overview easy to read and remember. Check your completed overview with your teacher.

90 100 110 120 1 3₀ 14 0 150 160 170 180 80 70 60 50 40 30 20 10 0 90 80 70 60 50 4 0 30 2 0 10 0 100 110 120 130 140 150 160 170 180 Pr_otr actor Co-interior

ANGLES

1 Draw labelled diagrams of these angles:

a ∠BKT b ∠FPR c angle MZQ

2 Use a protractor to measure each angle you drew in Question 1. Name the smallest angle and the largest angle.

3 Use a protractor to draw these angles.

a ∠JUG = 84° b ∠QRA = 117° c ∠POT = 41°

d ∠DGE = 150° e ∠SAR = 96° f ∠XDW = 210°

g ∠MNB = 195° h ∠PLO = 270° I ∠AMP = 300°

4 Write the name of each of these angles. Then label each one as acute, obtuse, right, reflex or straight.

5 Without measuring, find the size of each angle shown by a letter:

Chapter 2

Review

6 Label the marked angles as alternate, co-interior or corresponding:

7 Find the size of each angle shown with a letter:

g h i 82° t° 105° 25° p° q° r° x° x° x° Ex 2-12 a b c d e f x x x x Ex 2-12 y° g h i x° 37° z° 62° _m° p° 112° t° d° a° a b c a° 115° m° 35° k° 65° 130° q° 62° x° d° 125° d e f

8 Without measuring, find the size of each angle shown with a letter:

9 In each diagram below, is AB parallel to CD? Give a reason for your answer each time.

10 Draw a neat diagram to illustrate each of the following:

a an acute angle b supplementary angles

c a straight angle d vertically opposite angles

e alternate angles f an obtuse angle

g corresponding angles h a reflex angle i complementary angles j co-interior angles

Ex 2-12 x° 130° y° x° 64° m° 70° a° z° c° 38° 57° x° y° 145° z° a° 38° c° a b c d e Ex 2-13 a b c A C D B 45° 135° 110° 112° B D C A A C D B 74° 74° E F G H E F G H E F G H