Download Parallel and perpendicular lines

measurement anD Geometry • geometriC reasoning
4
5
Substitute the value of y into the equation.
Solve for x.
x + 120è = 180è
x = 180è - 120è
= 60è
Perpendicular lines
■
■
■
X
Perpendicular means ‘at right angles’.
When two line segments meet at right angles they are said
to be perpendicular. That is, the angle between them is 90è.
In the diagram at right, the angles ±AOX and ±XOB are each
right angled which means that the line segments AB and OX
are perpendicular.
To write ‘the line segments AB and OX are perpendicular’
using mathematical symbols, simply write AB ^ OX.
The symbol ^ means ‘is perpendicular to’.
A
O
B
rememBer
1. (a) Vertically opposite angles are equal in size.
(b) Adjacent angles are next to each other.
2. Corresponding (F) angles are equal in size.
3. Co-interior (C) angles add to 180è.
4. Alternate (Z) angles are equal in size.
5. When looking for the F, C or Z shape, remember that they may be backwards or
upside down.
6. The symbol for parallel is ||.
7. Supplementary angles add to 180è.
8. Complementary angles add to 90è.
9. If the given angles are in none of the above relations, we might need to find some
other angle first. This other angle must be related to both given angles.
10. Perpendicular means ‘at right angles’.
eXerCise
5F
individual
PatHWays
eBook plus
Activity 5-F-1
Parallel and
perpendicular lines
doc-1752
Parallel and perpendicular lines
FluenCY
1 Copy and complete this table.
Diagram
Type of angle relation
Associated shape
Corresponding
Rule
Are equal in size.
Activity 5-F-2
More parallel and
perpendicular lines
doc-1753
Alternate
Z
Activity 5-F-3
Advanced parallel and
perpendicular lines
doc-1754
eBook plus
Digital doc
Investigate parallel and
perpendicular lines
doc-6456
168
maths Quest 7 for the australian Curriculum
Add up to 180è.
measurement AND geometry • geometric reasoning
2 aCopy the diagram into your workbook. Clearly draw the ‘F’ shape on
your diagram and label the angle corresponding to the one that is
marked.
b Copy the diagram into your workbook. Clearly draw the ‘Z’ shape
on your diagram and label the angle alternate to the marked angle.
c Copy the diagram and label the angle co-interior to the marked angle.
Clearly draw the ‘C’ shape on your diagram.
3 Match each diagram with the appropriate name from the four options listed.
Diagram
Name
a
A Co-interior angles (C)
b
B Corresponding angles (F)
c
C Alternate angles (Z)
d
D None of the above
4 In the diagram at right, list all pairs of:
a vertically opposite angles
b corresponding angles
c co-interior angles
d alternate angles.
a b
c d
e f
g h
5 MC In the diagram below:
a Which angle is vertically opposite to angle p?
a k
b m
c r
d q
e t
r t
n p
k m
j q
Chapter 5 Geometry
169
measurement AND geometry • geometric reasoning
b Which angle is corresponding to angle p?
a k
b m
c r
d q
E t
c r
d q
E n
c r
d q
E n
c Which angle is co-interior to angle p?
a k
b m
d Which angle is alternate to angle p?
a k
b m
e Name two angles adjacent to angle k.
6 WE 19 For each of the following diagrams:
i state the type of angle relationship
ii find the value of the pronumeral.
a
b
p
c
45°
q
60°
s
65°
d
e
f
132°
72°
m 70°
t
n
7 Find the value of the pronumerals in each of the following diagrams, giving reasons.
a
b
116° 64°
c
62°
b
38°
z
y
d
e
68°
72° g
44°
44°
f
135°
k
h
120°
110°
8 a WE 20 Find the value of the pronumerals in each of the following diagrams.
i
ii
C
A
123°
B
G
H
D
D
E
E
b
F
170
Maths Quest 7 for the Australian Curriculum
iii
A
x
B
A
B
C
F
137°
y
G
H
G
H
F
62°
E
C
D
measurement AND geometry • geometric reasoning
iv
v
A
F
z z
E
M
N
P
H
80°
G
vi
P
B
p
V
J
C
U
T
W
X
160°
Q
R
S
Y
q
V
T
D
q
W
Z
Q
R
S
b Name all the perpendicular lines in the diagrams above.
understanding
9 If the angle allied to x is 135è, find the size of angle x.
p
10 If the angle corresponding to y is 55è, find the size of angle y.
11 A hill is at an angle of 30è to the horizontal. A fence is put in, consisting
of a railing parallel to the ground and vertical fence posts. Find the angle
between the top of the fence post and the rail.
12 Two gates consist of vertical posts, horizontal struts and diagonal
beams. Find the angle, a, as shown in the gates below.
a
30°
b
a
40°
a
50°
Reasoning
13 Is the line AB parallel to line CD? Explain your answer.
A
B
48°
84°
133°
C
D
14 a What shapes can you identify in this picture?
Make a list.
b Name three places in the picture where you see
parallel lines.
When examining a photograph of a 3-dimensional
object, you notice that some sets of parallel lines
(such as the wires in the railing) appear to meet in
the distance. This point is called the vanishing point
and is used by architects, engineers and artists to
create diagrams and artworks in perspective.
c Identify the vanishing point in the photograph.
Chapter 5 Geometry
171
measurement anD Geometry • geometriC reasoning
15 Perspective drawing also allows us to create the illusion of looking
at the same object from a number of different angles. For example,
consider a simple box as shown at right.
The two figures below are views of the cube from different
angles. In all cases the top of the box has been shaded.
Figure A
Figure B
a Find the vanishing point in each diagram, by extending the edges in the diagram until
they meet.
b True or false?
i The viewer is above the box in Figure A.
ii The viewer is behind the box in Figure B.
iii The viewer of Figure A is probably taller than the viewer of Figure B.
eBook plus
Digital docs
WorkSHEET 5.2
doc-6618
172
maths Quest 7 for the australian Curriculum
reFleCtion
Can you think of a way to
remember the different types of
angles associated with parallel
lines and their special properties?