5. Angles, 2D and 3D

166

In this chapter you will learn about:• vertically opposite angles• angles in a quadrilateral• finding missing angles in an angle problem • building 3D solids and drawing views of 3D solids• drawing 3D solids on isometric paper

5.1 Vertically Opposite AnglesGet Ready

You will need:

• ruler • protractor

Exercise 5.1

Warm Up

1 In your notebooks, copy and complete.

a 45 + 45 = ? b 90 = 2 × ?

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167167

c 180 – ? = 90 d 4 × 90 = ? e ? × 60 = 180

f 360 – 90 = ? g 135 = 180 – ?

2 What am I? Choose the correct name from the box below. Write the answers in your notebook.

a

I join two points and I can make the side of a shape.

b

Where two lines meet, they make an ? .

c

I have four equal sides and four equal angles.

d

I have three straight sides.

e

I have four sides and four equal angles.

f

I am another name for 90˚.

rectangle

angle

right angle

square

line

triangle

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+5

95 100 180

P Q

S R H I

F

G

A B

D C F

G

E

D

D E

FG

This side is called DE

This side is called FG

Word fact: Opposite

means on the other side.

3 Solve. Use a number line to help you.

a 95 + ? = 180

b 30 + ? = 180 c 180 – 27 = ?

d 131 + ? = 180 e 180 – 64 = ?

Side DE is the opposite side FG.

4 Name the opposite side.

a AB b PS c HI d EG

5 Copy and complete.

a 20 + ? = 90 b 15 + ? = 90 c ? + 65 = 90

d 105 + ? = 180 e 30 + 50 + 90 + 10 = ?

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HANDS ONACtiVITY

6 Measure.

a b

c

d

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7 a Look at angles p, q, r, and s. Estimate the size of each angle.

b Measure the angles. Use your protractor.

c What do you call angle s?

d Add the angles on line AB.

e In your notebook, draw your own line CD. Draw three angles on CD. Mark your angles x, y, and z.

f Measure your angles x, y, and z.

g Add the angles on line CD.

h Copy and complete the rule:

Angles on a straight line add up to ? .

8 Work out angles a, b, and c. Do not use a protractor.

–35 –30 –25 –20

–34 –23

129°a 73°c

62°b

Hint 129° + ? = 180°

?

129 180

Ap

q

r

s

B

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9 Work out angles a, b, and c.

10 Work out angles a, b, and c.

a

12 12

3

456

78

9

1011

c

12 12

3

456

78

9

1011

b

12 12

3

456

78

9

1011

33°

89°

a

43°

67°b 60° 60°

c

Hint Angles in a triangle add up to 180°.

Use a number line to help you add and subtract.

Hint Angles in a full turn add up to 360°.

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VertexVertex

Vertex

jk lm

ab

cd

gh

if

Main Exercise

11 Measure all the angles in each sketch.

a b c

12 What do you notice about the angles in Q11?

Vertically opposite angles are equal.

Vertically opposite angles are found on intersecting lines.

These angles are not vertically opposite:

Hint Use a protractor.

TALKABOUT IT

Word fact: The word for more than one

vertex is vertices.

Math fact: Vertically opposite

angles are ‘opposite a vertex’.

Vertically oppositeangles

Vertically oppositeangles

Vertically oppositeangles

Vertically oppositeangles

Vertically oppositeangles

Vertically oppositeangles

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13 Are a and b vertically opposite angles? Write yes or no in your notebook.

14 Work out the size of x. Write your answers in your notebook.

15 Work out the sizes of the unknown angles.

ab

ab a

b

a

b

ab a b

a b

ab

32°95°

x127° x x

a

a

a

b

b

c

c

b c d

hgfe

155°25°

pq 140°

40°rq

168°

12°

m

n

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16 Here is a pattern on a gate in Abu Dhabi. Which angles are vertically opposite?

17 Light bends when it passes from one material to another. This diagram shows light passing through a glass lens. Work out m and n. Write the answers in your notebook.

ab

ce

d

f

30°28°

mn

TALKABOUT IT

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18 Explaining The diagram shows two straight lines crossing.

a Amal says, ‘Angle a is 68˚.’

Is she correct? Explain your answer.

b What is the size of angle b?

19 Work out the size of x and y.

b Work out the sizes of p and q.

c Work out the sizes of a, b and x.

ab112°

x

y65°

45°

Hint x = ? – 65˚ (Angles on a line add up to 180˚).

Which angles are vertically opposite x?

pq

66°

47°

b

a x60°

54°

Hint First use vertically opposite angles.

The angles in a triangle add up to 180°.

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d Work out the sizes of angles c, d, and e.

e Work out the sizes of angles a, b, c, and x.

e

c d62°

b

ax

c64°

68°

Hint Angle c = angle d.

(Base angles of an isosceles triangle are equal.)

THINKABOUT IT

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20 Identifying Patterns Work in pairs.

You will need:

• popsicle sticks

Person 1: Place two popsicle sticks so they cross over each other and make four angles.

Person 2: Use a protractor to measure only one angle.

Work out the other three angles.

Person 1: Use a protractor to measure the other three angles and check your partner’s answers.

Repeat, changing how you place the popsicle sticks.

HANDS ONACtiVITY

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5.2 Angles in a QuadrilateralGet Ready

You will need:

• scrap paper • tracing paper

Exercise 5.2

Warm Up

1 In your notebooks, copy and complete each sequence.

a 45, 90, 135, 180, ? , ? , ? , 360

b 30, 60, 90, ? , ? , ?

c 180, 175, 170, ? , ? , ?

d 360, 315, 270, ? , ? , ? , ? , ?

2 Work out:

a 360 ÷ 4 b 90 ÷ 2

3 Copy and complete these rules:

a Angles in a triangle add up to ? .

b Angles on a straight line add up to ? .

c Vertically opposite angles are ? .

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4 a Work out the angles a, b and c.

b Add the three angles in the sketch. What is the total?

5 These lines intersect at point P. They make a right angle.

Add the four angles at point P.

6 a Measure the angles at point Q.

b Add the angles at point Q.

c Draw your own five angles around a point.

d Measure and add the angles.

7 Look at your answers to Q4–6.

Copy and complete this rule:

Angles at a point add up to ? .

ab

c

54°

Word fact: When lines cross, we say they

intersect.

Hint Use angles on a line

to work out a. Use vertically opposite angles to work out b and c.

a bcd

efQ

P

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Main Exercise

8 Identifying Relationships What is the same and different about these quadrilaterals?

TALKABOUT IT

Word fact: A quadrilateral is a shape with four straight sides and four vertices.

A

B

C

D

E F

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AB

CD

9

a Trace quadrilateral A from Q8.

b Cut it out. c Tear off the four corners.

d Arrange them around a point on a piece of paper.

e What do you notice?

f Repeat this exercise with any three quadrilaterals from Q8.

10 Here is a quadrilateral cut in two:

a What two shapes does it make?

b What do the three red angles add up to?

c What do the three blue angles add up to?

HANDS ONACtiVITY

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11 Suggesting Conclusions Work with a partner. Talk about your answers to Q9–10. Copy and complete this rule:

Angles in a quadrilateral add up to ? .

12 Use a protractor. Measure the angles inside each quadrilateral.

Check they add up to 360˚.

TALKABOUT IT

HANDS ONACtiVITY

A B

C

E

D

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110°

80°

50°

n

13 Without using a protractor, work out the missing angles.

These sketches are not to scale.

a b

c d

14 Without using a protractor, work out the missing angles.

These sketches are not to scale.

a b

Hint x = 360° − ? − ? − ?

90° 70°

Q

RS

P125°

x60°

120°

U

VW

T

q

60°

75°

25°25°

z

A B

D C

y

W

65°

65°

Y

XZ xx

Hint 2x + 65 + 65 = 360

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W 88°

85°75° Y

X

Z

r

c

15 This is the back of a chair. Work out the missing angles.

16 Work out the sizes of angles p, q, r and s.

Hint The base angles of an

isosceles triangle are equal.

Hint Use vertically opposite angles.

80°

80° a

b c

d e

100°100°

43°

111°

q

s

rp

THINKABOUT IT

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17 Look at this pattern.

a What shapes are the blocks of color?

b Which angle is missing from the green block?

18 Generating Solutions With a partner.

Ahmed marks six dots on a circle.

He joins four dots to make a quadrilateral.

What different quadrilaterals can Ahmed make?

a Draw each quadrilateral. b Name it.

c Work out its angles.

45°

right angles

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5.3 Angle ProblemsExercise 5.3

Warm Up

1 Write each group of letters in alphabetical order:

a q p z x e c b

r s y a f

2 Look at this diagram. Name:

a Two angles on a straight line.

b Two angles that are vertically opposite.

c Angles in a triangle.

d Angles in a quadrilateral.

e Angles around a point.

3 Compare your answers to Q2 with a partner. Did you get the same answers? Are you both correct?

Word fact: Alphabetical order means the order that letters appear in the English alphabet

a

b

c

de

hf

i

j

g

TALKABOUT IT

b c

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4 Match:

a Angles in a triangle A 360˚

b Angles on a straight line B 90˚

c Angles at a point C 180˚

d Angles in a quadrilateral D 360˚

e Right angle E 180˚

5 a Write two sentences about this triangle.

Use your sentences to help you find:

b Angle p c Angle q

6 Hamda and Khawla look at this diagram.

Hamda writes x = 89°.

She gives the reason: Vertically opposite angles are equal.

Khawla writes x = 91°.

She gives the reason: Angles on a straight line add up to 180°.

a Who is correct: Hamda or Khawla?

b How did the reasons help you to work out who was correct?

c Work out y. Give a reason.

writeABOUT IT

q

p

70°

xy

89°

Hint What type of triangle is this?

What do you know about its angles?

TALKABOUT IT

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Example

Work out the missing angles. Give reasons.

Answer

x = 180˚ − 55˚

x = 25˚ (angles on a line add up to 180˚).

y = 55˚ (vertically opposite angles are equal).

7 Work out the missing angles. Give reasons.

a

b

c

d

x y

55°

a b

48°

Hint Which angles are on a

straight line?

Hint Find angles in

alphabetical order.

Hint What shape is: ?

What do angles in this shape add up to?

p

q 57°

70°

y x100°

80°84°

e

f

g

108°

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ab

x

z y

80°

45°

p

q

m

75°

y 65°

a

b

8 In summer, the sun is higher in the sky than in winter. Work out the missing angles for each diagram. Give reasons.

a Summer b Winter

9 a Describe to a partner:

• Vertically opposite angles

• Angles on a line

• Angles in a triangle

• Angles in a quadrilateral.

b Describe the angles in each of these diagrams:

x

y

Sun

Earth

123° a

b

Sun

152°

Earth

TALKABOUT IT

TALKABOUT IT

Hint Use a sketch to help you.

Hint a and b are ? angles.

Math fact: Angles in a triangle add up to 180° can also be written: ∠s in Δ = 180°.

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10 Here is Aldar HQ in Abu Dhabi.

a What quadrilaterals are in the circle?

b Making Decisions Write down four angles that could be inside the quadrilateral.

c How did you choose your angles? Compare your answers with a partner. Did you get the same answers? Are you both correct?

11 Work out the missing angles. Give reasons.

a

Hint What do the angles add

up to? Which angles should be the same?

Hint Find angles in

alphabetical order. For each answer give one of these reasons:

∠s on a straight line = 180°

∠s in Δ = 180°.

ps

r q 115°48°

TALKABOUT IT

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a

b

cde

f g

108°46°

58°

48°

b c

d e

12 a Work out the angles of b Work out all the missing angles. this roof.

13 Use software such as Microsoft Word, Power Point or iPhoto. Make a summary about these kinds of angles:

• Angles in a quadrilateral

• Angles on a line

• Angles in a triangle

• Vertically opposite angles.

m

k l72°

28° x

y z

78°

a

c

b

146°75°

n

m

q

p

101°

73°66°

Something wrong with this brief!I’ve copied it for shape but that 73° angle must be wrongly labeled.Otherwise the picture would be ten times as high!I’ve switched p and 73°?

a b

c

60°

eLearningeLearning

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5.4 Building and Drawing 3D SolidsGet Ready

You will need:

• multilink cubes • red, yellow, and blue stickers

• whiteboard marker • isometric paper

• ruler • squared paper • sharp pencil

Exercise 5.4

Warm Up

1 How many cubes make each solid?

Word fact: A solid is a 3D object such as a cube.

3D is short for three-dimensional. This means that something has length, width and height.

a b

c d e

TALKABOUT IT

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2 Is the flag on top, on the side or in front?

a b c

3 Which faces are blue – top, right or front?

a b

c d e

Main Exercise

4 Build these solids. Keep them all on your desk for later.

a b c

d e

Word fact: Top

Side

Front

Word fact: A face is a

flat surface of a solid.

HANDS ONACtiVITY

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5 Look at the cubes in each solid from Q4. Do not turn the solids around.

a Put a red sticker on all the top cube faces you can see.

b Put a blue sticker on all the side cube faces you can see.

c Put a yellow sticker on all the front cube faces you can see.

6 How many different solids can you build with 4 cubes?

Hint A red sticker on one top cube face:

A yellow sticker on one front cube face:

A blue sticker on one side cube face:

THINKABOUT IT

HANDS ONACtiVITY

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Isometric Drawings

7 Draw these patterns on isometric paper.

a b c

8 Find each pattern from Q1 on the isometric drawings below.

9 Use the patterns from Q7 to help you draw a single isometric cube like this:

TALKABOUT IT

HANDS ONACtiVITY

Hint Start with the top of the solid.

Word fact: Isometric paper has dots at equal intervals. It is used to draw solid shapes.

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10 Organizing Information Draw these solids on isometric paper.

11 Organizing Information Build these solids. Then draw them on isometric paper.

a height = 5 cubes, width = 1 cube, length = 1 cube

b height = 2 cubes, width = 3 cubes, length = 3 cubes

12 Organizing Information Draw these solids on isometric paper.

a b c

d

Hint Start with the top

of the solid. Then draw vertical lines.

Hint

Length = 2 cubes Width = 1 cubes

Height = 4 cubes

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Different Views

13 Here is a police car:

Which arrow shows the:

a Front view? b Side view? c Top view?

14 For each view, write top view, side view or front view.

a b c

Word fact: A view is what you see from one position.

B

C

A

Hint Use the arrows in Q1 to help you.

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15 Look at the drawings of solids below. Build each solid with multilink blocks.

a b c d

16 Use stickers to mark the faces:

• red on top

• blue on the side

• yellow on the front.

Example

Draw the top, side and front view of this solid.

Answer

The yellow faces are in front. Front view:

The blue faces are on the side. Side view:

The red faces are on top. Top view:

HANDS ONACtiVITY

Hint There is a color hint in

the drawings.

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17 Draw the top, side and front view of each solid from Q15.

18 Here is a Rubik’s Cube.

a How many small cubes make the Rubik’s cube?

b How many squares on one face?

c Organizing Information Choose one view. Draw it. Color all the small squares correctly.

19 Organizing Information Draw the top view of each solid.

a b c

d e

Hint Look at the solids you built.

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20 Build these solids.

Top view Side view Front view

Solid A

Solid B

21 Use the video feature on a digital camera or smartphone. Make a video to show the front, back and side views of a 3D solid you have built.

THINKABOUT IT

Hint Each of the solids above is cuboid in shape.

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Summary

Vertically opposite angles are opposite each other where two lines intersect.

Vertically opposite angles are equal.

The angles inside a quadrilateral add up to 360˚.

Angle rules like these let us calculate missing angles. We do not need to use a protractor.

A solid has length, width and height.

Top, front, and side views show how a solid looks from different positions.

Top, front, and side views are 2D drawings made from squares. They look like this:

We use isometric paper to make 3D drawings of solids. A cube looks like this:

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x41°a

b

137°

x

x

32°

Review

1 What is the relationship between:

a angles a and c?

b angles b and d?

c angles b and c?

2 Work out the size of x in each sketch. Explain each answer.

a b

3 Vertically opposite angles:

A are right angles B add up to 180˚ C are equal

4 Angles on a line add up to:

A 360˚ B 180˚ C 90˚

5 Angles in a quadrilateral add up to:

A 360˚ B 180˚ C 90˚

6 Work out the size of the missing angles.

a b

a

b

c

d

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95°

150°

x

y

z100°

60°

65°

q

s r

p

25°

62°

d

b

ca

108°

82° 78°

g

f

82°

70°

98°

a

27°

240°20°b

7 Work out the sizes of the missing angles in these quadrilaterals.

a b

8 Work out the missing angles. Explain all the relationships you use.

a b

c d

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9 How many cubes are needed to make each solid?

a b

10 Draw the top, side and front view of each solid from Q9. The front is the shaded side.

11 Draw the following solids on isometric paper.

a a cube

b a solid built from three cubes in a row

12 Draw this solid on isometric paper.

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I can statementsWhat can you do?

• I can identify vertically opposite angles.

• I can work out missing angles where two lines intersect.

• I can work out missing angles in a triangle or quadrilateral.

• I can find the missing angle in an angle problem.

• I can give reasons when I find missing angles.

• I can build 3D solids.

• I can draw different views of 3D solids.

• I can draw 3D solids on isometric paper.

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