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2.2 Angles and proof
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, definitions and
derived properties
Why learn this?
Angles can be crucia
lly important
in some extreme spor
ts.
interior angle
Interior and exterior angles add up to 180°. Level 5
exterior angle
The interior angles in a triangle sum to 180°. Level 5
The interior angles in a quadrilateral sum to 360°. Level 6
b
The exterior angle of a triangle is equal to the sum of the
two interior opposite angles. For example, a ⴙ b ⴝ e. Level 6
a
e
Did you know?
A convention is an accepted mathematical way to show some information. Level 7
A definition is a precise description. For example, the definition of a square is: a shape with
exactly four equal sides and four equal angles. Level 7
The word ‘angle’
comes from
the Latin word
A derived property is information that can be worked out from a definition.
For example, each angle of a square is 90° because they sum to 360° and are all equal. Level 7 ‘angulus’, which
means ‘a corner’.
Level 5
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°
80°
I can use
interior and
exterior angles to
calculate angles
p
q
52°
Work out the size of angle p.
a Copy and complete these sentences to identify the interior and exterior angles.
E
A
D
U
Z
T
Tip
S
117°
C
105°
B
EBC is
X
B
98°
84°
C
Y
W
V
W
X
an interior angle.
i YZW is
.
ii SWX is
iii XTU is
.
A
The marked angle
is ABC or BCA.
.
b Calculate the missing angles marked on the diagrams.
Calculate the size of the lettered angles, stating any angle facts that you use.
a
b
c
d
94°
d
107°
p
70°
68°
22
Get in line
x y
s
t
110°
58°
u
convention
q
105°
definition
64°
98° e
derived property
Level 6
I can calculate
interior and
exterior angles
of triangles and
quadrilaterals
exterior angle
Sketch this diagram.
x
b
Then copy and complete these sentences.
a Angle x is equal to angle a because they are
b Angle y is equal to angle
xⴙbⴙyⴝ
because they are
because they lie on a
c Since x ⴝ a and y ⴝ
,
aⴙbⴙcⴝ
ⴙbⴙ
angles.
a
y
c
angles.
.
.
Angles in a triangle
sum to 180°. Angles
in a quadrilateral
sum to 360°.
.
Sketch this diagram.
The quadrilateral has been split into two triangles.
a ⴙ b ⴙ c ⴝ 180°
Continue the proof to show that angles in a
quadrilateral sum to 360°.
b
c
e
a
d
b
Then copy and complete this proof.
triangle sum to
b cⴙxⴝ
c
x
.
because they lie on a
c So a ⴙ b ⴙ c ⴝ c ⴙ
So a ⴙ b ⴝ x.
a
I can follow
a proof that the
sum of angles in a
quadrilateral is 360°
f
Sketch this diagram.
ⴝ 180° because angles in a
I can follow a
proof that the
sum of angles in a
triangle is 180°
Learn this
This proves that angles in a triangle sum to
a aⴙbⴙ
Level 6
.
I can follow a
proof that the
exterior angle of a
triangle is equal to
the sum of the two
interior opposite
angles
.
Decide whether each statement is a definition, a convention or a derived property.
Angles on a straight line sum to 180°.
An interior angle is an angle inside a shape.
Derived property
Definition
a The exterior angle of a triangle is equal to the sum of the two interior
opposite angles.
Level 7
I can recognise
the difference
between conventions,
definitions and
derived properties
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.
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.
interior angle
proof
quadrilateral
triangle
2.2 Angles and proof
23
2.3 Constructing triangles
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
Why learn this?
Triangles are a stron
g shape
used in the constructi
on of
many bridges.
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
hypotenuse
The hypotenuse of a right-angled triangle is the
longest side and is opposite the right angle. Level 6
Did you know?
The word
‘triangle’ is made
up of ‘tri’, which
means ‘three’ and
‘angle’. A triangle
has three angles.
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
Level 5
Draw these angles accurately using a ruler and protractor.
Label each angle as reflex or obtuse.
a 138°
b 294°
Construct these triangles using a
ruler and protractor.
c 197°
a
I can use a
protractor to
draw obtuse and
reflex angles to the
nearest degree
d 176°
b
B
4 cm
X
I can construct
a triangle given
two sides and the
included angle (SAS)
6 cm
72°
A
Make an accurate drawing
of these triangles.
a
C
6 cm
4 cm Z
b
B
45°
Y
85°
5 cm
73°
I can construct a
triangle given
two angles and the
included side (ASA)
60° 3 cm
C
A
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 find how
much wood is needed for one truss.
24
Get in line
acute
88° 65°
5m
angle
compasses
These triangles are all
right-angled triangles.
Which letter marks the
hypotenuse of each
triangle?
c
a
b
c
e
a
b
d
h
d
g
Level 6
l
I can identify
the hypotenuse
in a right-angled
triangle
k
i
j
f
I can construct
a triangle given
three sides
Use compasses and a ruler to construct a triangle with sides AB ⴝ 7 cm,
AC ⴝ 6 cm and BC ⴝ 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.
I can use a ruler
and compasses
to construct the
perpendicular from
a point on a line
segment
A
4 cm
5 cm
Level 7
A motor cycle stunt man is building a ramp so he can jump over four cars.
Here is the side-view of his ramp.
I can construct a
right-angled
triangle if I know
the lengths of the
hypotenuse and
another side (RHS)
10 m
8m
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.
Watch out!
4m
a Draw an accurate scale drawing of the ladder against the wall.
b Use your drawing to find how far the ladder reaches up the wall.
Don’t rub out
your construc tion lines
as they show that you
have used the compasses
correctly.
1m
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.
Hint: Try to construct the triangle first.
A
B
Drawing triangles 1
Drawing triangles 2
1 Draw a triangle and label the vertices
A, B and C.
1 Draw a right-angled triangle using
compasses and a ruler.
2 Measure the sides AB and AC.
3 Measure the angle BAC.
2 Measure the hypotenuse and one of
the other sides.
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.
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.
5 Check your partner’s triangle with the
original.
4 Check your partner’s triangle with the
original.
construction
hypotenuse
perpendicular
obtuse
2.3 Constructing triangles
25
2.4 Special quadrilaterals
Know the properties of quadrilaterals
Solve geometrical problems involving quadrilaterals and
explain the reasons
Why learn this?
Many buildings are
made of
rectangles and othe
r quadrilaterals.
How many different
shapes can you
see in this photo?
Quadrilateral properties:
rectangle
square
parallelogram
rhombus
isosceles trapezium
kite
arrowhead
Level 6
When solving problems using the properties of shapes it is important to explain
your reasoning. Level 6 & Level 7
Level 6
I can identify
angle, side
and symmetry
properties of simple
quadrilaterals
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
Tip
Number of lines of symmetry
0
1
2
Some cells may
contain more than
one shape.
4
0
1
2
Did you know?
a rectangle
b square
c parallelogram
d rhombus
e kite
f arrowhead
g isosceles trapezium
The prefix ‘quadri-’ comes from
the Latin word for four. Can
you think of any other words
that begin with ‘quad’?
Draw a rectangle and cut it out.
I can identify
and begin to
use angle, side and
symmetry properties
of quadrilaterals
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.
26
Get in line
arrowhead
isosceles trapezium
kite
parallelogram
Level 6
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.
b
c
30°
In a rhombus, one of the angles is 40°.
Work out the sizes of the other angles.
x
40°
The square and rhombus
are quadrilaterals with
equal length sides.
T
Look at this arrowhead.
TSV ⴝ 45°, STV ⴝ 30°
I can solve
geometrical
problems using
properties of triangles
and quadrilaterals
30°
b TVU
c SVU
45°
V
S
In this rectangle, calculate angle EBD.
Show your steps for solving this problem and
explain your reasoning.
U
A
B
72°
b ABE
D
A
Work out the sizes of these angles.
Explain your reasoning.
B
b c
a
c CBE
B
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.
quadrilateral
rectangle
rhombus
C
Level 7
C
d
I can use
reasoning to
solve more complex
geometrical problems
d BCD
F
A
I can use
reasoning to
solve geometrical
problems
56°
E
a FAB
I can solve simple
geometrical
problems using
properties of
quadrilaterals
Learn this
y
z
Calculate
a TUV
a
I can identify
and begin to
use angle, side and
symmetry properties
of quadrilaterals
70°
E
65°
D
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.
square
symmetry
2.4 Special quadrilaterals
27
2.5 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
Why learn this?
Understanding
perpendicular lines
can help
you appreciate their
use in
Lines that meet at right angles are perpendicular. Perpendicular
buildings and on ro
ads.
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
Level 6
I can construct
the mid-point
and perpendicular
bisector of a line
segment
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.
10 m
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?
I can name the
parts of a circle
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.
Learn this
a an acute angle of your choice
‘Bisect’ means to cut
something into two
equal parts.
b an angle of 90° drawn with a protractor
c an obtuse angle of your choice
28
Get in line
I can construct
the bisector of an
angle
arc
bisector (bisect)
chord
circle
compasses
I can construct
the perpendicular
from a point to a line
segment
Tip
Copy the diagram.
Using compasses and a ruler,
draw the perpendicular at X.
Check after
you have drawn a
perpendicular line to see if
it looks to be at a right
angle.
X
3 cm
Level 6
A
Make a copy of this diagram.
Construct the perpendicular from
point A to the line.
5 cm
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.
S
Level 7
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?
A
I can explain
how standard
constructions
using a ruler and
compasses relate to
the properties of two
intersecting circles
with equal radii
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.
diameter
perpendicular
radius
right angle
tangent
2.5 More constructions
29
2.6 Angles in polygons
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
Why learn this?
polygons to solve problems
Polygons are found
in many
places in nature. Whe
n lava
cools it can form colum
ns in
the shape of polygon
s.
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 ⴚ 2) ⴛ 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
sum of interior angles
The interior angle of a regular polygon ⴝ _________________________. Level 6
number of sides
You can use interior and exterior angles in polygons to solve problems. Level 7
Level 6
Explain how you calculate the sum of the interior angles in
I can explain
how to find the
interior angle sum of a
polygon
a a quadrilateral
b a pentagon.
I can explain
how to calculate
the interior and
exterior angles of
regular polygons
a Explain how you find the size of an interior angle in
a regular pentagon.
b Explain how you find the size of an exterior angle in
a regular pentagon.
I can calculate
the sums of the
interior and exterior
angles of irregular
polygons
a What is the sum of the interior angles in
i 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
I ⴙ E ⴝ 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?
30
Get in line
exterior angle
hexagon
interior angle
irregular polygon
Level 6
a Calculate the exterior angle of a regular hexagon.
I can calculate
the interior and
exterior angles of
regular polygons
b Calculate the size of each interior angle in a regular hexagon.
Copy and complete this table.
Regular
polygon
Number of
sides
equilateral
triangle
3
square
4
regular
pentagon
5
Sum of interior Size of each Sum of exterior Size of each
angles
interior angle
angles
exterior angle
180°
360°
regular
hexagon
regular
octagon
a How do you find 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.
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.
Level 7
Learn this
b The interior angle of a regular polygon is 156°.
How many sides does the polygon have?
The exterior angles
of a polygon always
add up to 360°.
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 of
a BCD
b CDA
B
C
A
H
D
I can solve
harder problems
using properties of
angles, parallel and
intersecting lines, and
triangles and other
polygons
E
67.5°
G
c ADH
I can use the
interior and
exterior angles of
regular polygons to
solve problems
F
Did you know?
A
Polygon poster
Make a poster of all the facts you know about the interior and
exterior angles of polygons.
B
Polygons are used to create
complex-shaped computer
graphics. Next time you play
a computer game, see how
many polygons you can spot.
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?
octagon
pentagon
quadrilateral
regular polygon
triangle
2.6 Angles in polygons
31