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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
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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 problemsWork 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 propertiesUse 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°.
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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
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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 11 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 21 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
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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
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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 ParallelogramsDraw 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 quadrailateralsA 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
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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
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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 circles1 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 circles1 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
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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 posterMake a poster of all the facts you know about
the interior and exterior angles of polygons.
B Tessellating polygonsInvestigate 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.