points, lines and surfaces · basic elements of geometry 15 a surface can be either flat or curved....
TRANSCRIPT
![Page 1: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/1.jpg)
Basic Elements of Geometry
Parallel Lines and Perpendicular Lines
1
Let’
s h
ave
a M
ath
Ch
at
Points, Lines and Surfaces
![Page 2: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/2.jpg)
Basic Elements of Geometry
2
What is geometry?
![Page 3: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/3.jpg)
Basic Elements of Geometry
3
Geometry is an old branch of mathematics which
studies shapes and sizes of figures.
Points, lines and surfaces are the three basic
elements of geometry.
![Page 4: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/4.jpg)
Basic Elements of Geometry
4
What is a point?
![Page 5: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/5.jpg)
Basic Elements of Geometry
5
A point which has no size,
represents a position in space or on a plane.
Label a point by a capital letter.
Use a small cross ‘×’ or a small dot ‘’
to represent a point.
e.g. × A B
We call it ‘point A’. We call it ‘point B’.
![Page 6: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/6.jpg)
Basic Elements of Geometry
6
What is a line?
![Page 7: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/7.jpg)
Basic Elements of Geometry
7
A line which has no width,
is made up of an infinite number of
points and can extend indefinitely.
Use a letter (e.g. L or l) to label a line.
Lines can either be straight line or curves.
e.g. L
l
straight line curve
![Page 8: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/8.jpg)
Basic Elements of Geometry
8
We can also take any two points on a
straight line and label the straight line
by those two points.
e.g.
M N × ×
line MN or line NM
![Page 9: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/9.jpg)
Basic Elements of Geometry
9
What is a line
segment?
![Page 10: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/10.jpg)
Basic Elements of Geometry
10
A line segment is a part of a straight line
and has two end-points.
e.g.
X Y × ×
line segment XY or
line segment YX
Label a line segment by its end-point.
![Page 11: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/11.jpg)
Basic Elements of Geometry
11
The length of a line segment is the
distance between the end-points.
Sometimes, we can use the name of the
line segment to denote its length.
e.g.
X Y × ×
The length of the line segment XY is 8 cm,
or we can simply write XY = 8 cm.
![Page 12: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/12.jpg)
P R
T
Q
S
Basic Elements of Geometry
12
In the figure, line segments PQ
and RS and drawn.
(a) Mark a point U on PQ and draw the line segment RU.
(b) Name the longest line segment in the figure.
U
PQ
![Page 13: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/13.jpg)
P R
T
Q
S
Basic Elements of Geometry
13
The point at which two straight lines intersect is
called the point of intersection.
Points of intersection can also be made by curves.
T is the point of
intersection of PQ and RS.
![Page 14: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/14.jpg)
Basic Elements of Geometry
14
What is a surface?
![Page 15: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/15.jpg)
Basic Elements of Geometry
15
A surface can be either flat or curved.
A flat surface is called a plane.
When a surface is curved, such kind of
surface is a curved surface.
e.g.
A plane has no thickness
and contains infinite number
of points and lines.
plane
curved
surface
A plane can extend endlessly.
We can label a plane by a
Greek letter π.
π π
![Page 16: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/16.jpg)
Parallel Lines and Perpendicular Lines
16
Now I am going to introduce parallel lines and
perpendicular lines.
A
B C
D
E
F
![Page 17: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/17.jpg)
X Y
V Z
Parallel Lines and Perpendicular Lines
17
Parallel lines are straight lines on the same plane
which never intersect each other no matter how
far they are extended.
e.g.
Use the symbol ‘//’ to represent ‘is parallel to’.
XY // VZ, XV // YZ
Same number of arrowheads (e.g. ‘’ or ‘ ’)
on lines in the same direction are used to
indicate that the lines are parallel.
![Page 18: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/18.jpg)
Parallel Lines and Perpendicular Lines
18
Perpendicular lines are two straight lines
which intersect and form a right angle.
e.g.
Use the symbol ‘’ to represent
‘is perpendicular to’.
The mark of a right angle ‘ ’ is used to indicate that the
two lines are perpendicular.
M
O
N
P
M
O
N
P
MN OP MN OP
![Page 19: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/19.jpg)
Parallel Lines and Perpendicular Lines
19
The distance between two parallel
lines is always the same.
e.g.
parallel
lines
distance
![Page 20: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/20.jpg)
B
D
C
E
A
F
G
Parallel Lines and Perpendicular Lines
20
AB // GF, BC // FE
(a) Mark parallel lines in the figure with arrowheads.
Example 1
(b) Write down all pairs of parallel lines in the figure.
![Page 21: Points, Lines and Surfaces · Basic Elements of Geometry 15 A surface can be either flat or curved. A flat surface is called a plane. When a surface is curved, such kind of surface](https://reader034.vdocument.in/reader034/viewer/2022042414/5f2f100e1e61f90c9352cb6e/html5/thumbnails/21.jpg)
Parallel Lines and Perpendicular Lines
21
Example 2
Write down all pairs of perpendicular lines in the figure.
L1
L2
L3 L4 L5
L1 L3, L2 L4