22

�� Identify interior and exterior angles in triangles and quadrilaterals�� Calculate interior and exterior angles of triangles and quadrilaterals�� Understand the idea of proof�� Recognise the difference between conventions, defi nitions and

derived properties

2.2 Angles and proof 2.2 Angles and proof

�� Interior and exterior angles add up to 180°. Level 5

�� The interior angles in a triangle sum to 180°. Level 5Level 5

�� The interior angles in a quadrilateral sum to 360°. Level 6

�� The exterior angle of a triangle is equal to the sum of the two interior opposite angles. For example, a � b � e. Level 6

�� A convention is an accepted mathematical way to show some information. Level 7

�� A defi nition is a precise description. For example, the defi nition of a square is: a shape with exactly four equal sides and four equal angles. Level 7

�� A derived property is information that can be worked out from a defi nition.For example, each angle of a square is 90° because they sum to 360° and are all equal. Level 7

convention definition derived property exterior angle

interior angle

exterior angle

a

b

e

Level 5 I can use

interior and exterior angles to calculate angles

Get in line

Angles can be crucially important in some extreme sports.

Why learn this?

Work out the size of angle q.q is an interior angle.q and 52° lie on a straight line, so they sum to 180°.q = 180 – 52 = 128°

Work out the size of angle p.

a Copy and complete these sentences to identify the interior and exterior angles.

� EBC is an interior angle.

i �YZW is . ii �SWX is . iii � XTU is .

b Calculate the missing angles marked on the diagrams.

qp80°

52°

105°

84° 98°

117°

BC Y W V W X

U T S

X

ZDE

A

Calculate the size of the lettered angles, stating any angle facts that you use.

x y s t

p d

equ68°

94°70°

105°

58°64°

110°

98°

107°a b c dLevel 6

I can calculate interior and exterior angles of triangles and quadrilaterals

Did you know?Did you know?The word ‘angle’ comes from the Latin word ‘angulus’, which means ‘a corner’.

TipTip

The marked angle is � ABC or � BCA.

B

C

A

23interior angle proof quadrilateral triangle

Sketch this diagram.

Then copy and complete these sentences.

a Angle x is equal to angle a because they are angles.

b Angle y is equal to angle because they are angles.

x � b � y � because they lie on a .

c Since x � a and y � , a � b � c � � b � .

This proves that angles in a triangle sum to .

Sketch this diagram.The quadrilateral has been split into two triangles.

a � b � c � 180°Continue the proof to show that angles in a quadrilateral sum to 360°.

Sketch this diagram.

Then copy and complete this proof.

a a � b � � 180° because angles in a

triangle sum to .

b c � x � because they lie on a .

c So a � b � c � c � .So a � b � x.

Level 6 I can follow a

proof that the sum of angles in a triangle is 180°

I can follow a proof that the sum of angles in a quadrilateral is 360°

I can follow a proof that the exterior angle of a triangle is equal to the sum of the two interior opposite angles

a c

bx y

a c

b

x

Level 7 I can recognise

the difference between conventions, defi nitions and derived properties

Decide whether each statement is a defi nition, a convention or a derived property.

Angles on a straight line sum to 180°. Derived property

An interior angle is an angle inside a shape. Defi nition

a The exterior angle of a triangle is equal to the sum of the two interior opposite angles.

b The dashes on opposite sides of a rectangle show that the sides are the same length.

c A triangle has three sides and three interior angles.

d Parallel lines are marked with arrows pointing in the same direction.

2.2 Angles and proof

b

ad

ec

f

A Angle problems

Work with a partner. Each draw a triangle with the interior and exterior angles marked.Tell your partner two of the interior angles from your triangle. Challenge them to work out the other interior angle and the exterior angles.Check their answers to see if they are correct.

B Triangle properties

Use a dynamic geometry program to construct a triangle with a line going through one vertex that is parallel to the opposite side.Drag any of the vertices to explore what happens.

Learn thisLearn thisAngles in a triangle sum to 180°. Angles in a quadrilateral sum to 360°.

24

�� Draw an angle accurately using a protractor�� Construct a triangle using a protractor and a ruler�� Construct a triangle using compasses and a ruler�� Draw a right-angled triangle using compasses and a ruler

2.3 Constructing triangles2.3 Constructing triangles

�� You can construct a triangle using a ruler and a protractor if you know either two sides and the included angle (SAS) or two angles and the included side (ASA). Level 5

�� You can construct a triangle using a ruler and compasses if you know the length of all three sides (SSS). Level 6

�� The hypotenuse of a right-angled triangle is the longest side and is opposite the right angle. Level 6

�� Lines that meet at right angles are perpendicular. Perpendicular lines can be constructed using compasses. Level 6

�� You can construct a right-angled triangle using a ruler and compasses if you know the lengths of the hypotenuse and one of the shorter sides (RHS). Level 7

acute angle compasses

Triangles are a strong shape used in the construction of many bridges.

Why learn this?

Draw these angles accurately using a ruler and protractor.Label each angle as refl ex or obtuse.

a 138° b 294° c 197° d 176°

Construct these triangles using a ruler and protractor.

Make an accurate drawing of these triangles.

An architect is calculating the length of wood required to make trussels for a roof.The width of the roof is 5 m and the two angles to the horizontal are 88° and 65°.

a Using a scale of 1 cm represents 1 m, draw an accurate scale drawing of the roof.

b Measure the length of each sloping beam to fi nd how much wood is needed for one truss.

Level 5 I can use a

protractor to draw obtuse and refl ex angles to the nearest degree

I can construct a triangle given two sides and the included angle (SAS)

B

A

Xa b

YZ6 cm

6 cm4 cm

4 cmC72°

5 cm

3 cm73°

85°

60°

45°

B

A

C

a b

5 m65°88°

Did you know?Did you know?The word ‘triangle’ is made up of ‘tri’, which means ‘three’ and ‘angle’. A triangle has three angles.

I can construct a triangle given two angles and the included side (ASA)

Get in line

hypotenuse

25construction hypotenuse perpendicular obtuse

These triangles are all right-angled triangles.Which letter marks the hypotenuse of each triangle?

Use compasses and a ruler to construct a triangle with sides AB � 7 cm, AC � 6 cm and BC � 5 cm.

Construct a triangle with sides of length 9 cm, 7 cm and 8 cm using compasses and a ruler.

Using compasses and a ruler, draw the perpendicular to the line at point A.

Level 6 I can identify

the hypotenuse in a right-angled triangle

I can construct a triangle given three sides

I can use a ruler and compasses to construct the perpendicular from a point on a line segment

b

f

d ig

hl

k

j

ea

ca b c d

4 cm 5 cm

A

A motor cycle stunt man is building a ramp so he can jump over four cars.Here is the side-view of his ramp.

a Draw an accurate scale drawing of the ramp using a ruler and compasses.

b What is the height of the top of the ramp?

A 4 m ladder leans against a wall with its base 1 m from the wall.

a Draw an accurate scale drawing of the ladder against the wall.

b Use your drawing to fi nd how far the ladder reaches up the wall.

Mark wants to construct a triangle with sides of length 5 cm, 3 cm and 9 cm.Explain why Mark’s triangle is impossible to construct.

Level 7 I can construct a

right-angled triangle if I know the lengths of the hypotenuse and another side (RHS)

8 m

10 m

A Drawing triangles 1

1 Draw a triangle and label the vertices A, B and C.

2 Measure the sides AB and AC.

3 Measure the angle BAC.

4 Describe the triangle to your partner by telling them the information about the two sides and the angle. Your partner draws the triangle you have described.

5 Check your partner’s triangle with the original.

B Drawing triangles 2

1 Draw a right-angled triangle using compasses and a ruler.

2 Measure the hypotenuse and one of the other sides.

3 Describe the triangle to your partner by telling them the information about the two sides and the angle. Your partner draws the triangle you have described.

4 Check your partner’s triangle with the original.

2.3 Constructing triangles

Don’t rub out your construction lines

as they show that you

have used the compasses correctly.

Watch out!Watch out!

Hint:Hint: Try to construct the triangle fi rst.

1 m

4 m

26

2.4 Special quadrilaterals 2.4 Special quadrilaterals�� Know the properties of quadrilaterals�� Solve geometrical problems involving quadrilaterals and

explain the reasons

�� Quadrilateral properties:

Level 6

�� When solving problems using the properties of shapes it is important to explain your reasoning. Level 6 & Level 7

Many buildings are made of rectangles and other quadrilaterals. How many different shapes can you see in this photo?

Why learn this?

kite arrowheadsquarerectangle rhombusparallelogram isosceles trapezium

arrowhead isosceles trapezium kite parallelogram

Nathaniel said ‘A square is a rectangle’. Is this true? Explain your answer.

Which of these statements are always true for a rectangle?A All its sides are equal.B It has four lines of symmetry.C It has four right angles.

Copy this table. Complete it by writing each shape name in the correct position.

Number of pairs of parallel sides

Number of lines of symmetry0 1 2 4

012

a rectangle b square c parallelogram

d rhombus e kite f arrowhead

g isosceles trapezium

Draw a rectangle and cut it out.

a Cut along one of the diagonals. Rearrange the pieces to make another quadrilateral.

b Write the name of the new quadrilateral that you have made.

c Write one geometrical fact about this shape.

Level 6

TipTipSome cells may

contain more than one shape.

Did you know?Did you know?The prefi x ‘quadri-’ comes from the Latin word for four. Can you think of any other words that begin with ‘quad’?

Get in line

I can identify angle, side and symmetry properties of simple quadrilaterals

I can identify and begin to use angle, side and symmetry properties of quadrilaterals

27quadrilateral rectangle rhombus square symmetry

Sketch an equilateral triangle in one of its sides.

a Write the name of the quadrilateral that is formed.

b Which of these statements are always true for this special quadrilateral?A The diagonals bisect at right angles.B The angles are all equal.C It has two pairs of parallel sides.D It has four lines of symmetry.

Look at this rectangle. One of the diagonals is drawn.Work out the sizes of angles angles a, b and c.

In a rhombus, one of the angles is 40°.Work out the sizes of the other angles.

Look at this arrowhead.�TSV � 45°, �STV � 30°

Calculatea �TUV b �TVU c �SVU

In this rectangle, calculate angle EBD. Show your steps for solving this problem and explain your reasoning.

Level 6

30° a

bc

40°

x

z

y

I can solve geometrical problems using properties of triangles and quadrilaterals

V

S U

T

30°

45°

I can use reasoning to solve geometrical problems

72°56°

A B

E D C

Work out the sizes of these angles. Explain your reasoning.

a �FAB b �ABE c �CBE d �BCD

Level 7 I can use

reasoning to solve more complex geometrical problems

70° 65°

dba c

F E D

A B C

2.4 Special quadrilaterals

Learn thisLearn thisThe square and rhombus are quadrilaterals with equal length sides.

B Parallelograms

Draw a parallelogram like this.Label three angles with their sizes.Challenge your partner to work out the missing angles and explain their reasons.Use what you know about the properties of parallelograms to check their answers.

A Special quadrailaterals

A game for two players. Each secretly draw a special quadrilateral. Take turns to tell each other one property of your shape. Try to guess each other’s shape. Score 1 point if you guess correctly from one property, 2 points from two properties, and so on. The player with the lowest score wins.

I can identify and begin to use angle, side and symmetry properties of quadrilaterals

I can solve simple geometrical problems using properties of quadrilaterals

28

2.5 More constructions2.5 More constructions�� Know the names of parts of a circle�� Use a straight edge and compasses to construct the

perpendicular bisector of a line and an angle, and the perpendicular to a line

�� Use a straight edge and compasses to investigate the properties of overlapping circles

�� Lines that meet at right angles are perpendicular. Perpendicular lines can be constructed using compasses. Level 6

�� The angle bisector cuts the angle in half. The perpendicular bisector cuts the line in half at right angles. Both can be constructed using compasses. Level 6

�� The perpendicular from a point to a line segment is the shortest distance to the line. Level 6

�� When the points of intersection of two identical overlapping circles are joined to the centres, a rhombus is formed. Level 7

�� A right-angled triangle can be constructed using a ruler and compasses if you know the length of the hypotenuse and one of the shorter sides. Level 7

arc bisector (bisect) chord circle compasses

Using only a ruler and compasses, draw the perpendicular bisectors of these line segments. Mark the mid-point of each line segment.

a a straight line segment AB of length 6 cm

b a straight line segment BD of length 8 cm

A construction company is building two houses, 10 m apart. The architect’s plans look like this.

a Copy the plan, using a scale of 1 cm to represent 1 m.

b Construct the perpendicular bisector of the 10 m line:

c A fence will be built on the perpendicular bisector.What can you say about the position of the fence?

Copy this circle with radius 4 cm. Add these labels.

a radius b diameter c chord

d arc e tangent f circumference

Use compasses and a ruler to draw the bisector of these angles.

a an acute angle of your choice

b an angle of 90° drawn with a protractor

c an obtuse angle of your choice

Level 6

I can name the parts of a circle

I can construct the mid-point and perpendicular bisector of a line segment

I can construct the bisector of an angle

Learn thisLearn this‘Bisect’ means to cut something into two equal parts.

Understanding perpendicular lines can help you appreciate their use in buildings and on roads.

Why learn this?

Get in line

10 m

29 diameter perpendicular radius right angle tangent

Make a copy of this diagram.Construct the perpendicular from point A to the line.

Copy the diagram. Using compasses and a ruler, draw the perpendicular at X.

A construction company is building a bridge across a river.Copy the diagram and draw the perpendicular from point S across the river to show where the bridge should be built.

Level 6

3 cm 5 cm

X

A

a Using compasses, draw two circles of radius 4 cm that overlap.

b Join the centres of the circles with a straight line and draw the chord that is common to both circles.

c Join the centres of the circles the points where the circles intersect.What do you notice about the quadrilateral that is formed?

Level 7 I can explain

how standard constructions using a ruler and compasses relate to the properties of two intersecting circles with equal radii

A Triangles in circles

1 Draw a circle, using compasses or dynamic geometry software.

2 Mark three points on the circumference of the circle.

3 Join up these points to make a triangle.

4 Construct the perpendicular bisector of each side of your triangle.

5 What do you notice?

6 What happens when the vertices of the triangle are moved to different points on the circumference?

B Polygons in circles

1 Draw a circle, using compasses or dynamic geometry software.

2 Mark four points on the circumference of the circle.

3 Join up these points to make a quadrilateral.

4 Construct the perpendicular bisector of each side of your quadrilateral.

5 What do you notice?

6 Investigate other polygons inside a circle.

2.5 More constructions

TipTipCheck after

you have drawn a

perpendicular line to see if

it looks to be at a right angle.

S

I can construct the perpendicular from a point to a line segment

30

�� Find the sum of the interior and exterior angles of polygons�� Find an interior and exterior angle of a regular polygon�� Use the interior and exterior angles of regular and irregular

polygons to solve problems

�� An interior angle and its corresponding exterior angle sum to 180°. Level 5

�� The sum of the interior angles in an n-sided polygon is (n � 2) � 180°. Level 6

�� The sum of the exterior angles in any polygon is always 360°. Level 6

�� A regular polygon has all sides of equal length and all angles equal. Level 6

�� The interior angle of a regular polygon � sum of interior angles

_________________________ number of sides

. Level 6

�� You can use interior and exterior angles in polygons to solve problems. Level 7

2.6 Angles in polygons2.6 Angles in polygons

exterior angle hexagon interior angle irregular polygon

Polygons are found in many places in nature. When lava cools it can form columns in the shape of polygons.

Why learn this?

Explain how you calculate the sum of the interior angles in

a a quadrilateral

b a pentagon.

a Explain how you fi nd the size of an interior angle in a regular pentagon.

b Explain how you fi nd the size of an exterior angle in a regular pentagon.

a What is the sum of the interior angles ini a quadrilateral ii a pentagon iii a hexagon?

b Calculate the sum of the interior angles in a 10-sided polygon.

Look at this quadrilateral.At each vertex the sum of the interior and exterior angles is 180°.

I � E � 180°Explain why this is true.

a Draw a quadrilateral with the exterior angles marked, like the one in Q4.

b Use a protractor to measure each exterior angle.Find the sum of the exterior angles.

c Repeat parts a and b for a pentagon and a hexagon.

d What do you notice about the sum of the exterior angles of a polygon?

Level 6 I can explain

how to fi nd the interior angle sum of a polygon

I can explain how to calculate the interior and exterior angles of regular polygons

I can calculate the sums of the interior and exterior angles of irregular polygons

IE

Get in line

31octagon pentagon quadrilateral regular polygon triangle

Level 6 I can calculate

the interior and exterior angles of regular polygons

a The exterior angle of a regular polygon is 18°.ii How many sides does the polygon have?ii Calculate the size of each interior angle.

b The interior angle of a regular polygon is 156°. How many sides does the polygon have?

It is possible to draw a polygon that has interior angles that sum to 1300°?Explain your reasoning

The diagram shows a regular octagon.The line BC is parallel to the line AD.

Calculate the size ofa �BCD

b �CDA

c �ADH

Level 7 I can use the

interior and exterior angles of regular polygons to solve problems

I can solve harder problems using properties of angles, parallel and intersecting lines, and triangles and other polygons

CB

FG

A

H

D

67.5° E

2.6 Angles in polygons

A Polygon poster

Make a poster of all the facts you know about the interior and exterior angles of polygons.

B Tessellating polygons

Investigate which regular polygons tessellate. Look at the interior angles. How can you tell by looking at the interior angles whether a shape will tessellate? Why will a regular hexagon and a square tessellate?

Learn thisLearn thisThe exterior angles of a polygon always add up to 360°.

a Calculate the exterior angle of a regular hexagon.

b Calculate the size of each interior angle in a regular hexagon.

Copy and complete this table.

Regular polygon

Number of sides

Sum of interior angles

Size of each interior angle

Sum of exterior angles

Size of each exterior angle

equilateral triangle

3 180° 360°

square 4regular

pentagon5

regular hexagonregular octagon

a How do you fi nd the sum of the interior angles in an n-sided polygon?

b Calculate the size of the interior and exterior angles in a regular 16-sided shape.

Did you know?Did you know?Polygons are used to create complex-shaped computer graphics. Next time you play a computer game, see how many polygons you can spot.