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Key words
Angles
protractor
acute
obtuse
reflex
line segments
Distinguish between, and estimate acute, obtuse and reflex angles
Measure angles accurately
Know how to label lines, angles and shapes
An angle is a measure of turn and
is measured using a protractor .
C
B
The vertex B is at
the centre of the
protractor.
Read off the size of the
angle in degrees here.
Angle ABC ⫽ 25°.
A
The line segment BA is
on the 0° line of the protractor.
An acute angle is less than 90o. A right angle is exactly 90o.
An obtuse angle is between 90o and 180o.
A reflex angle is between 180o and 360o.
In maths there are agreed ways of labelling diagrams.
In this diagram the line segments AB and BC are labelled.
The blue lines show that AB is the same length as BC.
An angle can be described in different ways.
^C or B.
This diagram shows angle ABC or AB
A
B
C
D
When labelling a diagram, we label the vertices in clockwise
alphabetical order.
C
c This side is
DE, or c.
E
Example
Which is which? Look at this diagram and find:
a) an acute angle
b) an obtuse angle
c) a right angle
d) a pair of equal line segments.
R
P
S
a) QRS and STP are acute angles.
b) PQR and RST are obtuse angles.
c) TPQ is a right angle.
d) PQ and QR are equal.
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Exercise 7.1 ..........................................................................................
Which is which? Describe these angles correctly.
B
a) The angle coloured orange is _____.
b) The angle coloured blue is _____.
A
c) The angle coloured green is _____.
D
d) The angle coloured yellow is _____.
e) The angle coloured black is _____.
C
E
List all the marked angles in this diagram. For each one, state whether it is acute, a right
angle, obtuse or reflex.
A
C
B
D
G
F
E
Measure the lettered angles and then list them in order of size, smallest first.
a
b
d
c
e
f
Draw angles of 40°, 65°, 87°, 103°, 15°, 210° and 355°. Label them clearly.
Play with a partner. Take turns to draw an angle. Both of you write down an estimate for
the size of the angle, then measure it. Whoever is the closer to the estimate wins one
point. If you estimate exactly you get two points. Repeat several times to find the overall
winner.
Investigation
Draw a large triangle RST with no sides or angles the same. Mark the mid-point of
TR as U, the mid-point of RS as V and the mid-point of ST as W. Join UVW to make
another triangle. Mark the mid-points of the sides of UVW and join them. Repeat as
many times as you can. Write down what you notice about:
a) the lengths of the sides of the triangles
b) the angles of the triangles
c) the areas of the triangles.
Angles 69
07 Section 7 pp068-073.qxd
7.2
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Key words
Lines, angles and shapes
exterior
interior
Know that sum of angles at a point is 360º, and that the sum of angles
on a straight line is 180º
Understand that the angle sum of a triangle is 180º
Understand that the angle sum of a quadrilateral is 360º
Understand that the exterior angle of a triangle is equal to the sum of the two interior
opposite angles
a
a
a
b
b
c
d
a ⫹ b ⫹ c ⫹ d ⫽ 360°
a ⫹ b ⫽ 180°
c
b
d
a
c
b
a ⫹ b ⫹ c ⫽ 180°
a ⫹ b ⫹ c ⫹ d ⫽ 360°
Angles which meet at a point on a straight line add up to 180°.
Angles which meet at a point add up to 360°.
The three angles of any triangle add up to 180°.
The four angles of any quadrilateral add up to 360°.
a ⫹ x ⫽ 180°
(angles on a straight line)
a ⫹ b ⫹ c ⫽ 180°
(angles on a triangle)
so x ⫽ b ⫹ c
b
c
a x
The exterior angle of a triangle is made by extending one of the sides. The exterior
angle of a triangle is equal to the sum of the two opposite interior angles of the
triangle.
Example
Calculate the size of the lettered angles.
Give reasons for your answers.
W
b
Z
120°
40°
e
X
c
50°
d a
Y
a is 90° because it is the exterior angle of the triangle XYZ.
b and c are 30° because they are base angles of the triangle WXY.
d is 90° because angles in a quadrilateral add up to 360°.
e is 150° because angles on a straight line add up to 180°.
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ZXY 50° and YZX 40°
50 40 90°
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Exercise 7.2 ..............................................................................................
Calculate (don’t measure) the lettered angles.
a)
b)
c)
30°
40°
a
b
60°
d 95°
95° e
c
100°
d)
e)
60°
f)
70°
150°
70°
h
60°
f
20°
20°
g
Sketch a triangle PQR. Mark in PQ QR and PQR 80°. Calculate the sizes of the
other angles.
Sketch a quadrilateral ABCD. Mark in CAB 40°, BCA 45°, BCD 90° and
ADC 60°.
Calculate <ABC and <DAC.
Calculate (don’t measure) the lettered angles.
Remember that the base angles in an
isosceles triangle are equal.
Give reasons for your answers.
q
r
110°
30°
20°
f g
j
k
l
n
p
30°
70°
m
Measure the lettered angles.
Check that angles on a straight line add up
to 180° and that the angles in the triangle
add up to 180°.
What do you notice about:
a) angles g and j
b) angles c and f
j
g
e c
f
d
Investigation
A pentagon can be divided into three triangles, so all the
angles in a pentagon must add up to 3 180° 540°.
Draw polygons with six, seven, eight sides and so on.
Divide the polygons up into triangles and find what all
the angles add up to.
What if the polygons were regular polygons?
What would be the size of each of the angles?
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7.3
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Key words
Recognising equal angles
parallel
perpendicular
alternate
corresponding
vertically opposite
Recognise vertically opposite angles and angles on a straight line
Identify alternate angles on parallel lines
Identify corresponding angles on parallel lines
Parallel lines are straight lines which never ever meet or cross.
Perpendicular lines cross or meet each other at right angles (90°).
You can use a set square and ruler to draw parallel and perpendicular lines.
B
D
A
G
C
E
F
H
Perpendicular lines
Parallel lines
Parallel lines can produce alternate and corresponding angles.
c
a
b
a
d
b
c
d
Alternate angles are equal.
Corresponding angles are equal.
When straight lines intersect they produce
vertically opposite angles.
Vertically opposite angles are equal.
Example
Calculate the size of the lettered angles.
Give reasons for your answers.
a
d
c
b
60°
b
c d
40°
a
e
a is 30° because angles in a triangle add up to 180°.
b is 30° because it is an alternate angle to angle a.
c is 110° because angles in a triangle add up to 180°.
d is 70° because angles on a straight line add up to 180°.
e is 70° because it is a corresponding angle to angle d.
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Exercise 7.3 ..............................................................................................
Look at this diagram:
a)
b)
c)
d)
e)
f)
Which angle is an alternate angle to a?
Which angle is an alternate angle to b?
Which angle is a corresponding angle to c?
Which angle is a corresponding angle to d?
Which angle is vertically opposite e?
Which angle is vertically opposite f?
s
f
a r
e
p
m
c
b
d
There may be more than one answer.
Draw a rectangle EFGH where EF 5 cm and FG 3 cm.
Label the parallel and perpendicular sides.
Draw in the diagonals EG and FH and mark in GEF 40°.
Calculate the size of all the angles.
Remember to label the
vertices in order clockwise.
Calculate the size of the lettered angles.
Give reasons for your answers.
There are a number of reasons you might give.
Compare your answers and the reasons you
gave for them with a partner.
c
d
a
b
40°
Calculate the size of the lettered angles in the following diagrams:
i)
ii)
40°
a
j
h
60°
iii)
r
b
60°
40°
d
f
k
i
g
p
e
m
q s n
Investigation
Make up a question for your partner to answer. Draw a diagram including parallel
lines and triangles or quadrilaterals. Label all the vertices and any parallel or equal
length lines. Include five angles for your partner to find. Make sure that you have
included enough angle information for your partner to be able to calculate the size
of the missing angles.
Recognising equal angles 73